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| Introduction to Exponents (Indices) | Laws of Exponents (for Integer and Real Bases) | Powers with Negative Exponents |
| Uses of Exponents (Standard Form, Comparing Numbers) | ||
Exponents and Powers
Introduction to Exponents (Indices)
Understanding Repeated Multiplication
In mathematics, we often encounter situations where a number needs to be multiplied by itself multiple times. For example, consider the calculation $5 \times 5 \times 5$. Writing this out is manageable. However, if we need to multiply a number by itself 100 times, writing out the full expression becomes impractical and cumbersome, prone to errors, and difficult to read.
To address this, mathematicians developed a concise notation called exponents (also known as indices or powers). Exponents provide a shorthand way to represent repeated multiplication of the same number.
Definition of Exponents
An exponent is a small number (a superscript) written to the upper right of a base number. It tells us how many times the base number is used as a factor in a multiplication.
In the expression $a^n$:
- $a$ is the base: This is the number that is being multiplied by itself. The base can be any real number.
- $n$ is the exponent (or index or power): This is the number that indicates how many times the base is used as a factor in the multiplication. Initially, we focus on cases where $n$ is a positive integer, representing the count of factors.
The expression $a^n$ is read in various ways, such as "$a$ raised to the power of $n$", "$a$ to the $n$-th power", or simply "$a$ to the $n$".
The definition for a positive integer exponent $n$ is:
$\quad a^n = \underbrace{a \times a \times a \times \ldots \times a}_{\text{n factors}}$
[Definition of Exponent for positive integer n]
Here, the base $a$ is multiplied by itself $n$ times.
Examples of Exponential Notation
Let's look at some examples of how repeated multiplication is written using exponents and evaluate their values:
- $2^1$: The base is 2, the exponent is 1. 2 is used as a factor 1 time. $2^1 = 2$. (Any number to the power of 1 is itself).
- $2^2$: The base is 2, the exponent is 2. 2 is used as a factor 2 times. $2^2 = 2 \times 2 = 4$. (Read as "2 squared").
- $2^3$: The base is 2, the exponent is 3. 2 is used as a factor 3 times. $2^3 = 2 \times 2 \times 2 = 8$. (Read as "2 cubed").
- $2^5$: The base is 2, the exponent is 5. 2 is used as a factor 5 times. $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$.
- $10^1$: $10^1 = 10$.
- $10^2$: $10^2 = 10 \times 10 = 100$.
- $10^6$: $10^6 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1,000,000$. (Used in place value and scientific notation).
- $(-3)^2$: The base is -3, exponent is 2. $(-3)^2 = (-3) \times (-3) = 9$.
- $(-3)^3$: The base is -3, exponent is 3. $(-3)^3 = (-3) \times (-3) \times (-3) = 9 \times (-3) = -27$.
- $(\frac{1}{2})^4$: The base is $\frac{1}{2}$, exponent is 4. $(\frac{1}{2})^4 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1 \times 1 \times 1 \times 1}{2 \times 2 \times 2 \times 2} = \frac{1^4}{2^4} = \frac{1}{16}$.
- $(0.1)^2$: The base is 0.1, exponent is 2. $(0.1)^2 = 0.1 \times 0.1 = 0.01$.
Key Terms and Concepts
In the context of exponents, it's important to understand the specific terminology used:
- Base: The number that is repeatedly multiplied. It is written as the larger number. Example: In $a^n$, $a$ is the base.
- Exponent (Index/Power): The number written as a superscript that indicates the number of times the base is used as a factor. Example: In $a^n$, $n$ is the exponent.
- Exponential Form: A way of writing a number or expression using a base and an exponent, e.g., $2^5$.
- Expanded Form: Writing out the repeated multiplication that the exponential form represents, e.g., $2 \times 2 \times 2 \times 2 \times 2$.
- Value: The result obtained after performing the repeated multiplication, e.g., the value of $2^5$ is 32.
Special Cases of Positive Integer Exponents
Certain positive integer exponents have special names:
- Exponent of 1: Any non-zero number raised to the power of $1$ is equal to the base number itself. $a^1 = a$. This is consistent with the definition, as the base is used as a factor only once.
- Exponent of 2: Raising a number to the power of $2$ is called "squaring" the number. $a^2 = a \times a$. This term originates from geometry; the area of a square with side length 'a' is $a \times a = a^2$.
- Exponent of 3: Raising a number to the power of $3$ is called "cubing" the number. $a^3 = a \times a \times a$. This term also originates from geometry; the volume of a cube with side length 'a' is $a \times a \times a = a^3$.
The introduction to exponents as repeated multiplication for positive integer powers is the starting point. In subsequent sections, we will explore the rules that govern operations with exponents and how the definition of exponents is extended to include zero, negative, and rational values for the exponent.
Laws of Exponents (for Integer and Real Bases)
The laws of exponents, also known as the laws of indices or power rules, are a set of rules that significantly simplify calculations involving powers. These laws are derived directly from the definition of exponents as repeated multiplication and extend to various types of bases (including integers and real numbers) and exponents (including positive, zero, and negative integers, and later, rational and real numbers).
Let $a$ and $b$ be non-zero real numbers, and $m$ and $n$ be integers.
Laws of Exponents
Law 1: Product of Powers with the Same Base
When multiplying exponential expressions with the same base, add the exponents.
$\quad a^m \times a^n = a^{m+n}$
[Product Rule]
Derivation (for positive integer exponents $m$ and $n$):
By the definition of exponents:
$$ a^m = \underbrace{a \times a \times \ldots \times a}_{m \text{ times}} $$ $$ a^n = \underbrace{a \times a \times \ldots \times a}_{n \text{ times}} $$Multiplying $a^m$ by $a^n$:
$$ a^m \times a^n = \left(\underbrace{a \times a \times \ldots \times a}_{m \text{ times}}\right) \times \left(\underbrace{a \times a \times \ldots \times a}_{n \text{ times}}\right) $$This is a total of $m+n$ factors of $a$ being multiplied together:
$$ = \underbrace{a \times a \times \ldots \times a}_{m+n \text{ times}} $$By the definition of exponents, this is $a^{m+n}$.
$$ a^m \times a^n = a^{m+n} $$Example: $2^3 \times 2^4$. Using repeated multiplication: $(2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^7$. Using the law: $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$.
Law 2: Quotient of Powers with the Same Base
When dividing exponential expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator.
$\quad \frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0)$
[Quotient Rule]
Derivation (for positive integer exponents $m$ and $n$, with $m > n$):
By the definition of exponents:
$$ \frac{a^m}{a^n} = \frac{\overbrace{a \times a \times \ldots \times a}^{m \text{ factors}}}{\underbrace{a \times a \times \ldots \times a}_{n \text{ factors}}} $$We can cancel $n$ factors of $a$ from both the numerator and the denominator:
$$ = \frac{\overbrace{\cancel{a} \times \cancel{a} \times \ldots \times \cancel{a}}^{n} \times \overbrace{a \times a \times \ldots \times a}^{m-n}}{\underbrace{\cancel{a} \times \cancel{a} \times \ldots \times \cancel{a}}_{n}} = \underbrace{a \times a \times \ldots \times a}_{m-n \text{ factors}} $$By the definition of exponents, this is $a^{m-n}$.
$$ \frac{a^m}{a^n} = a^{m-n} $$Example: $\frac{3^5}{3^2}$. Using repeated multiplication: $\frac{3 \times 3 \times 3 \times 3 \times 3}{3 \times 3} = \frac{\cancel{3} \times \cancel{3} \times 3 \times 3 \times 3}{\cancel{3} \times \cancel{3}} = 3 \times 3 \times 3 = 3^3 = 27$. Using the law: $\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$.
This law is fundamental to defining zero and negative exponents, which is done in a way that maintains the consistency of this rule for all integer exponents.
Law 3: Power of a Power
When raising an exponential expression (a power) to another power, multiply the exponents.
$\quad (a^m)^n = a^{m \times n} = a^{mn}$
[Power of a Power Rule]
Derivation (for positive integer exponents $m$ and $n$):
By the definition of exponents, raising $a^m$ to the power of $n$ means multiplying $a^m$ by itself $n$ times:
$$ (a^m)^n = \underbrace{a^m \times a^m \times \ldots \times a^m}_{n \text{ factors}} $$Now, substitute the definition of $a^m$ for each factor:
$$ = \underbrace{(\underbrace{a \times \ldots \times a}_{m}) \times (\underbrace{a \times \ldots \times a}_{m}) \times \ldots \times (\underbrace{a \times \ldots \times a}_{m})}_{n \text{ groups of } m \text{ factors}} $$In total, there are $n$ groups, and each group contains $m$ factors of $a$. So, the total number of factors of $a$ is $m \times n$.
$$ = \underbrace{a \times a \times \ldots \times a}_{m \times n \text{ times}} $$By the definition of exponents, this is $a^{mn}$.
$$ (a^m)^n = a^{mn} $$Example: $(5^2)^3$. Using repeated multiplication: $(5^2)^3 = 5^2 \times 5^2 \times 5^2 = (5 \times 5) \times (5 \times 5) \times (5 \times 5) = 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^6 = 15625$. Using the law: $(5^2)^3 = 5^{2 \times 3} = 5^6 = 15625$.
Law 4: Power of a Product
When raising a product of two or more factors to a power, raise each factor to that power.
$\quad (ab)^n = a^n b^n$
[Power of a Product Rule]
Derivation (for positive integer exponent $n$):
By the definition of exponents, raising the product $(ab)$ to the power of $n$ means multiplying $(ab)$ by itself $n$ times:
$$ (ab)^n = \underbrace{(ab) \times (ab) \times \ldots \times (ab)}_{n \text{ factors}} $$Using the commutative and associative properties of multiplication, we can rearrange and group the factors:
$$ = \underbrace{(a \times a \times \ldots \times a)}_{n \text{ factors}} \times \underbrace{(b \times b \times \ldots \times b)}_{n \text{ factors}} $$By the definition of exponents, the first group is $a^n$ and the second group is $b^n$.
$$ = a^n b^n $$ $$ (ab)^n = a^n b^n $$Example: $(2 \times 3)^4$. Calculate the product first: $(2 \times 3)^4 = 6^4 = 6 \times 6 \times 6 \times 6 = 1296$. Using the law: $(2 \times 3)^4 = 2^4 \times 3^4 = (2 \times 2 \times 2 \times 2) \times (3 \times 3 \times 3 \times 3) = 16 \times 81 = 1296$.
Law 5: Power of a Quotient
When raising a quotient (a fraction) to a power, raise both the numerator and the denominator to that power.
$\quad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \neq 0)$
[Power of a Quotient Rule]
Derivation (for positive integer exponent $n$):
By the definition of exponents, raising the quotient $\frac{a}{b}$ to the power of $n$ means multiplying $\frac{a}{b}$ by itself $n$ times:
$$ \left(\frac{a}{b}\right)^n = \underbrace{\frac{a}{b} \times \frac{a}{b} \times \ldots \times \frac{a}{b}}_{n \text{ factors}} $$When multiplying fractions, we multiply the numerators together and the denominators together:
$$ = \frac{\overbrace{a \times a \times \ldots \times a}^{n \text{ factors}}}{\underbrace{b \times b \times \ldots \times b}_{n \text{ factors}}} $$By the definition of exponents, the numerator is $a^n$ and the denominator is $b^n$.
$$ = \frac{a^n}{b^n} $$ $$ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $$Example: $\left(\frac{2}{5}\right)^3$. Calculate the cube first: $\left(\frac{2}{5}\right)^3 = \frac{2}{5} \times \frac{2}{5} \times \frac{2}{5} = \frac{2 \times 2 \times 2}{5 \times 5 \times 5} = \frac{8}{125}$. Using the law: $\left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3} = \frac{8}{125}$.
Summary of Laws of Exponents
For non-zero real numbers $a, b$ and integer exponents $m, n$:
- Product Rule: $\quad a^m \times a^n = a^{m+n}$
- Quotient Rule: $\quad \frac{a^m}{a^n} = a^{m-n}$ (where $a \neq 0$)
- Power of a Power Rule: $\quad (a^m)^n = a^{mn}$
- Power of a Product Rule: $\quad (ab)^n = a^n b^n$
- Power of a Quotient Rule: $\quad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ (where $b \neq 0$)
These laws provide a foundation for manipulating and simplifying expressions involving exponents, applicable to any real number base (except where noted, such as the base being zero) and any integer exponent. The next section will discuss how the definition of exponents is extended to include zero and negative integer exponents while preserving these laws.
Powers with Negative Exponents
The initial definition of an exponent as repeated multiplication, $a^n = a \times a \times \ldots \times a$ (n times), is intuitive and works well when $n$ is a positive integer. However, for exponents to be useful in a wider range of mathematical contexts and algebraic manipulations, the definition needs to be extended to include the cases where the exponent is zero or a negative integer. These definitions are made in a way that ensures the existing laws of exponents remain valid.
Let $a$ be a non-zero real number.
Zero Exponent
How should we define $a^0$? We use the laws of exponents to guide us. Consider the Quotient of Powers law: $\frac{a^m}{a^n} = a^{m-n}$, where $a \neq 0$.
What happens if the numerator and denominator have the same exponent, i.e., $m = n$?
Applying the Quotient law:
$\quad \frac{a^m}{a^m} = a^{m-m} = a^0$
We also know that any non-zero quantity divided by itself is equal to $1$. So, if $a \neq 0$, then $\frac{a^m}{a^m} = 1$.
To maintain consistency with the Quotient of Powers law, we must define $a^0$ to be equal to $1$ for any non-zero base $a$.
$\quad a^0 = 1 \quad (\text{for } a \neq 0)$
[Definition of Zero Exponent]
The expression $0^0$ is generally considered an indeterminate form in basic algebra and is often left undefined. In some advanced contexts (like calculus for limits or set theory for counting functions), $0^0$ might be defined as 1 for convenience, but for general exponent rules with variable bases, the base must be non-zero for $a^0 = 1$.
Examples:
- $5^0 = 1$
- $(-7)^0 = 1$
- $(\frac{2}{3})^0 = 1$
- $(1,00,000)^0 = 1$
- $(\sqrt{2})^0 = 1$
Negative Exponents
How should we define $a^{-n}$ where $n$ is a positive integer? Again, we use the laws of exponents. Consider the Quotient of Powers law $\frac{a^m}{a^n} = a^{m-n}$ when the exponent in the denominator is larger than the exponent in the numerator ($n > m$).
Let's look at an example: $\frac{a^2}{a^5}$, where $a \neq 0$.
Using the definition of exponents and cancellation:
$$ \frac{a^2}{a^5} = \frac{a \times a}{a \times a \times a \times a \times a} = \frac{\cancel{a} \times \cancel{a}}{\cancel{a} \times \cancel{a} \times a \times a \times a} = \frac{1}{a \times a \times a} = \frac{1}{a^3} $$Using the Quotient of Powers law directly:
$$ \frac{a^2}{a^5} = a^{2-5} = a^{-3} $$To make the results consistent, it follows that $a^{-3}$ must be equal to $\frac{1}{a^3}$.
In general, for any positive integer $n$ and non-zero base $a$, we define $a^{-n}$ as the reciprocal of $a^n$.
$\quad a^{-n} = \frac{1}{a^n} \quad (\text{for } a \neq 0)$
[Definition of Negative Exponent]
From this definition, we can also see that $a^n = \frac{1}{a^{-n}}$. A term with a negative exponent in the numerator is equivalent to its reciprocal with a positive exponent in the denominator, and vice versa.
A base of zero raised to a negative exponent ($0^{-n}$ where $n > 0$) is undefined because it would involve division by zero ($0^{-n} = \frac{1}{0^n} = \frac{1}{0}$, which is undefined).
Examples:
- $5^{-1} = \frac{1}{5^1} = \frac{1}{5}$
- $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
- $(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}$
- $(\frac{2}{3})^{-2}$: Using the definition, $(\frac{2}{3})^{-2} = \frac{1}{(\frac{2}{3})^2} = \frac{1}{\frac{4}{9}}$. Dividing by a fraction is multiplying by its reciprocal: $1 \div \frac{4}{9} = 1 \times \frac{9}{4} = \frac{9}{4}$. Note the pattern $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$.
- $\frac{1}{10^{-2}}$: Using the definition $\frac{1}{a^{-n}} = a^n$, $\frac{1}{10^{-2}} = 10^{-(-2)} = 10^2 = 100$.
Applying Laws of Exponents with Zero and Negative Exponents
The power of defining zero and negative exponents in this way is that the laws of exponents previously derived for positive integer exponents remain valid for all integer exponents (positive, zero, and negative). This allows us to use the same simple rules for a much wider range of exponential expressions.
Example 1. Simplify $(2^{-3})^2 \times 4^2 \times 8^{-1}$.
Answer:
The expression is $(2^{-3})^2 \times 4^2 \times 8^{-1}$. We can simplify this using the laws of exponents. It is helpful to express all bases as powers of the same prime number, in this case, 2 ($4=2^2$, $8=2^3$).
Rewrite the expression with base 2:
$\quad (2^{-3})^2 \times (2^2)^2 \times (2^3)^{-1}$
Apply Law 3 (Power of a Power: $(a^m)^n = a^{mn}$) to each term:
$\quad = 2^{(-3) \times 2} \times 2^{2 \times 2} \times 2^{3 \times (-1)}$
$\quad = 2^{-6} \times 2^4 \times 2^{-3}$
Apply Law 1 (Product of Powers with the Same Base: $a^m \times a^n = a^{m+n}$): Add all the exponents.
$\quad = 2^{-6 + 4 + (-3)}$
$\quad = 2^{-6 + 4 - 3}$
$\quad = 2^{-2 - 3}$
$\quad = 2^{-5}$
Use the definition of negative exponent ($a^{-n} = \frac{1}{a^n}$):
$\quad = \frac{1}{2^5} = \frac{1}{32}$
The simplified value is $\mathbf{\frac{1}{32}}$.
Example 2. Simplify $\frac{5^{-3} \times 10^2}{2^{-2} \times 5^0}$.
Answer:
The expression is $\frac{5^{-3} \times 10^2}{2^{-2} \times 5^0}$. Express all bases in terms of prime factors (2 and 5). $10 = 2 \times 5$.
$\quad \frac{5^{-3} \times (2 \times 5)^2}{2^{-2} \times 5^0}$
Apply Law 4 (Power of a Product: $(ab)^n = a^n b^n$) and the definition of zero exponent ($a^0 = 1$):
$\quad = \frac{5^{-3} \times 2^2 \times 5^2}{2^{-2} \times 1}$
$\quad = \frac{2^2 \times 5^{-3} \times 5^2}{2^{-2}}$
Apply Law 1 (Product of Powers: $a^m \times a^n = a^{m+n}$) for the powers of 5 in the numerator:
$\quad = \frac{2^2 \times 5^{-3+2}}{2^{-2}} = \frac{2^2 \times 5^{-1}}{2^{-2}}$
Apply Law 2 (Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$) for the powers of 2 and 5:
$\quad = 2^{2 - (-2)} \times 5^{-1}$
$\quad = 2^{2+2} \times 5^{-1} = 2^4 \times 5^{-1}$
Calculate $2^4$ and use the definition of negative exponent for $5^{-1}$:
$\quad = 16 \times \frac{1}{5} = \frac{16}{5}$
The simplified value is $\mathbf{\frac{16}{5}}$ (or $3.2$ as a decimal). Note that we could have also used the definition of negative exponents earlier, moving terms between numerator and denominator: $\frac{5^{-3} \times 10^2}{2^{-2} \times 5^0} = \frac{10^2 \times 2^2}{5^3 \times 1} = \frac{100 \times 4}{125 \times 1} = \frac{400}{125}$. Simplifying $\frac{400}{125}$ by dividing numerator and denominator by their GCD (which is 25) gives $\frac{16}{5}$.
The definitions of $a^0 = 1$ (for $a \neq 0$) and $a^{-n} = \frac{1}{a^n}$ (for $a \neq 0$) are consistent with the laws of exponents, making these laws applicable to all integer exponents and simplifying a wide range of algebraic expressions.
Uses of Exponents (Standard Form, Comparing Numbers)
Exponents provide a concise and efficient way to represent repeated multiplication. This capability is particularly valuable when dealing with numbers that are extremely large or extremely small. Using exponents, especially powers of 10, forms the basis for standard form (scientific notation), which simplifies the representation, comparison, and manipulation of numbers spanning vast ranges of magnitude.
Representation of Large and Small Numbers using Powers of 10
Our decimal system is based on powers of 10. Exponents make it easy to represent these place values and, consequently, very large and very small numbers.
- Positive integer exponents of 10: Represent 1 followed by $n$ zeros. These are used for large whole numbers.
$10^1 = 10$
$10^2 = 10 \times 10 = 100$
$10^3 = 10 \times 10 \times 10 = 1,000$ (One Thousand)
$10^6 = 1,000,000$ (One Million)
$10^7 = 1,00,00,000$ (One Crore in the Indian System)
$10^9 = 1,000,000,000$ (One Billion or One Arab in the Indian System)
And so on.
- Negative integer exponents of 10: Represent fractions with 1 as the numerator and a power of 10 as the denominator. These are used for small numbers (between 0 and 1).
$10^{-1} = \frac{1}{10^1} = \frac{1}{10} = 0.1$ (One Tenth)
$10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01$ (One Hundredth)
$10^{-3} = \frac{1}{10^3} = \frac{1}{1000} = 0.001$ (One Thousandth)
$10^{-6} = \frac{1}{10^6} = \frac{1}{1,000,000} = 0.000001$ (One Millionth)
And so on.
Exponents of 10 make it easy to express the magnitude (size) of numbers, especially very large or very small ones.
Standard Form (Scientific Notation)
Standard form, also widely known as scientific notation, is a standardized way of writing numbers, particularly very large or very small ones, using powers of 10. It provides a compact and consistent format.
A number is expressed in standard form as the product of a coefficient (a number between 1 and 10, inclusive of 1) and an integer power of 10.
The general form is $a \times 10^n$, where:
- $a$ is the coefficient (sometimes called the mantissa). It is a real number such that $1 \le |a| < 10$. This means the absolute value of $a$ must be greater than or equal to 1 and strictly less than 10. It is written with the decimal point immediately following the first non-zero digit.
- $10^n$ is the power of 10.
- $n$ is the exponent (or index). It is an integer (positive, negative, or zero) that indicates the order of magnitude of the number.
Converting a number to Standard Form:
To write any given number in standard form, follow these steps:
- Move the decimal point in the number until there is only one non-zero digit to the left of the decimal point. The resulting number is the coefficient $a$.
- Count the number of places the decimal point was moved. This count is the absolute value of the exponent $n$.
- Determine the sign of the exponent $n$:
- If the original number was $10$ or greater (a large number), the decimal point was moved to the left, and the exponent $n$ is positive.
- If the original number was between $0$ and $1$ (a small number), the decimal point was moved to the right, and the exponent $n$ is negative.
- If the original number was already between $1$ and $10$ (exclusive of 10), the decimal point was effectively moved 0 places, and the exponent $n$ is $0$.
- Write the number as the product of the coefficient $a$ and $10^n$.
Example: Write the speed of light, approximately $300,000,000 \text{ m/s}$, in standard form.
Move the decimal point from the end 8 places to the left to get 3.0. Number of places moved = 8. Original number is large, so exponent is positive. $300,000,000 = 3.0 \times 10^8$.
Example: Write the diameter of a hydrogen atom, approximately $0.000000000106 \text{ m}$, in standard form.
Move the decimal point from its position 10 places to the right to get 1.06. Number of places moved = 10. Original number is small, so exponent is negative. $0.000000000106 = 1.06 \times 10^{-10}$.
Example: Write $8.75$ in standard form.
The number is already between 1 and 10. The decimal point doesn't need to move. The coefficient is 8.75. The exponent is 0. $8.75 = 8.75 \times 10^0$.
Standard form makes it easy to read and compare the relative size (order of magnitude) of numbers.
Comparing Very Large and Very Small Numbers using Standard Form
One of the most significant uses of standard form is simplifying the comparison of numbers, especially those of vastly different magnitudes. Comparing $a \times 10^n$ and $b \times 10^m$ in standard form is straightforward and follows a systematic procedure.
To compare two numbers $a \times 10^n$ and $b \times 10^m$ in standard form (where $1 \le |a| < 10$ and $1 \le |b| < 10$):
- Compare the Exponents ($n$ and $m$): The primary factor determining which number is larger is the exponent of 10. The number with the larger exponent is the greater number. This is because a higher power of 10 signifies a larger order of magnitude. Remember the order of integers: $\ldots -2 < -1 < 0 < 1 < 2 \ldots$. A larger exponent means a larger number (e.g., $10^5 > 10^3$, $10^{-5} > 10^{-7}$).
If $n > m$, then $a \times 10^n > b \times 10^m$.
- If the exponents are equal ($n = m$): If the exponents of 10 are the same, the numbers have the same order of magnitude. The comparison is then determined by the coefficients $a$ and $b$. Since both coefficients are already written in a form where the decimal point is after the first non-zero digit (i.e., they are between 1 and 10 in absolute value), their values are easily compared using standard decimal comparison rules. The number with the larger coefficient (considering signs if they are negative) is the greater number.
If $n = m$ and $a > b$, then $a \times 10^n > b \times 10^m$.
If $n = m$ and $a < b$, then $a \times 10^n < b \times 10^m$.
If $n = m$ and $a = b$, then $a \times 10^n = b \times 10^m$.
Example 1. Compare the mass of the Earth ($5.972 \times 10^{24} \text{ kg}$) and the mass of the Moon ($7.346 \times 10^{22} \text{ kg}$).
Answer:
We are comparing $5.972 \times 10^{24}$ and $7.346 \times 10^{22}$. Both are in standard form.
Compare the exponents of $10$ first:
The exponent for Earth's mass is 24.
The exponent for Moon's mass is 22.
Comparing the exponents: $24 > 22$.
Since the exponent 24 is greater than the exponent 22, the number with the exponent 24 is greater.
Therefore, $5.972 \times 10^{24} > 7.346 \times 10^{22}$.
The mass of the Earth is greater than the mass of the Moon.
Example 2. Compare the diameter of a red blood cell ($7 \times 10^{-6}$ meters) and the diameter of a virus ($2.5 \times 10^{-8}$ meters).
Answer:
We are comparing $7 \times 10^{-6}$ and $2.5 \times 10^{-8}$. Both are in standard form.
Compare the exponents of $10$ first:
The exponent for the red blood cell diameter is -6.
The exponent for the virus diameter is -8.
Comparing the exponents: $-6 > -8$ (remember that a smaller negative number is greater).
Since the exponent -6 is greater than the exponent -8, the number with the exponent -6 is greater.
Therefore, $7 \times 10^{-6} > 2.5 \times 10^{-8}$.
The diameter of a red blood cell is greater than the diameter of a virus.
Example 3. Compare $4.7 \times 10^6$ and $4.07 \times 10^6$.
Answer:
We are comparing $4.7 \times 10^6$ and $4.07 \times 10^6$. Both are in standard form.
Compare the exponents of $10$ first:
The exponents are equal ($6 = 6$).
Since the exponents are equal, we compare the coefficients:
The coefficient for the first number is 4.7.
The coefficient for the second number is 4.07.
Compare the coefficients: $4.7$ and $4.07$. Using standard decimal comparison (or padding with zero: 4.70 vs 4.07), we see that $4.7 > 4.07$.
Since the coefficient 4.7 is greater than the coefficient 4.07, the number with the coefficient 4.7 is greater.
Therefore, $4.7 \times 10^6 > 4.07 \times 10^6$.
Standard form, leveraging exponents of 10, is an indispensable tool for representing and comparing numbers across the vast scales encountered in scientific and quantitative domains, making their magnitudes easily comprehensible and comparable.