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# Word Problems Involving Exponents Study Guide

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Updated on Apr 25, 2014

## Introduction to Word Problems Involving Exponents

Infinity is a floorless room without walls or ceilings.

—AUTHOR UNKNOWN

This lesson will explain the rules for exponents and the various forms of numbers. Scientific notation examples and word problems involving exponents will also be demonstrated.

## The Basics of Exponents

Exponents are a way of writing large and small numbers in a shortened way. In the expression 23, the number 2 is the base and the number 3 is the exponent. The exponent tells how many times the base is used as a factor, or how many times it should be multiplied. For example, the expression 2 to the third power is written as 23= 2 × 2 × 2 = 8. 23is the exponential form, 2 × 2 × 2 is the expanded form, and 8 is the standard form of this number.

#### Tip:

There are three useful forms of numbers when working with exponents. They are:

• Exponential form: 23
• Expanded form: 2 × 2 × 2
• Standard form: 8

## Exponent Rules

The rules of exponents can be explained by using the different forms of numbers.

### Multiplying

The expressions 45and 42have the same base. In expanded form, 45= 4 × 4 × 4 × 4 × 4 and 42= 4 × 4. To multiply them together, the expression becomes 45× 42= 4 × 4 × 4 × 4 × 4 × 4 × 4. This expanded form is multiplying seven 4s together; in other words, 47. This pattern can be explained as 45× 42= 45 + 2= 47. When multiplying with like bases, add the exponents.

### Dividing

To divide like bases, such as 34÷ 33, use expanded form and write division as a fraction to find a pattern. The expression becomes . Notice how the common factors in the numerator and denominator cancel. This pattern can be summarized as 34÷ 33= 34 – 3= 31= 3. When dividing like bases, subtract the exponents.

### Raising a Power to Another Power

When you are raising an exponent to a power, such as (24)3, the base becomes the expression within the parentheses. Therefore, to simplify this expression, rite the base 24as a factor three times. This is 24× 24× 24= 24 + 4 + 4= 212. When raising a power to another power, multiply the exponents.

#### Tip:

The rules for exponents can be summarized as the following:

• When multiplying like bases, add the exponents.
• x3× x4= x3 + 4 = x7
• When dividing like bases, subtract the exponents.
• When raising a power to another power, multiply the exponents.
• (x3)2= x3 × 2 = x6
• When adding or subtracting expressions with exponents, follow the order of operations.
• 32– 23 = 9 – 8 = 1

### Negative Exponents

A negative exponent tells you to take the reciprocal of the base. For example,. For expressions to be simplified, they should be written with positive exponents.

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### Related Questions

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