Simplifying Expressions with Exponents

Learning Objective

Introduction

Exponents can be attached to variables as well as numbers. When they are, the basic rules of exponentsThe value that indicates the number of times another value is multiplied by itself in exponential notation. The exponent, also called the power, is written in superscript. In the term `5^3`, `5` is the base and `3` is the exponent. and exponential notationA condensed way of expressing repeated multiplication of a value by itself. Exponential notation consists of a base and an exponent. In the exponential term `5^3`, `5` is the base and `3` is the exponent. It is a shorthand way of writing `5*5*5`. Also called exponential form. apply when writing and simplifying algebraic expressions that contain exponents.

Simple Rules of Exponents

Let’s look at some of the basic rules of exponents.

Any number or variable raised to a powerA way of describing the exponent in exponential notation. We can say the base is raised to the power of the exponent. For example, we can read `x^5` as “`x` raised to the power of `5`” or as “`x` raised to the `5`th power.” of one is simply that number. In return, any number or variable that doesn’t have an exponent shown with it can be considered to have an exponent of `1`. Below are some examples:

`5^1 = 5`

`18 = 18^1`

`x^1 = x`

`xy = x^1y^1`

Another rule of exponents is that any non-zero number or variable raised to a power of `0` equals `1.`  `x^0=1` for `x!=0`

Simplify the expression `x^0y^1z^2`

 

A) `0`

 

B) `xyz^2`

 

C) `yz^2`

 

D) `y^1z^2`

 

As with numbers, variables raised to a negative power are equivalent to `1` over the variable to the same but positive power. For example:

`x^-2=1/x^2`

`4y^-5=4*1/y^5=4/y^5`

Exponent of `0` or `1` and Negative Exponents

 

Any number or variable raised to a power of `1` is the number itself. `n^1=n`

 

Any non-zero number or variable raised to a power of `0` is equal to `1``n^0=1`

 

Any non-zero number `n` and any integer `x``n^-x=1/n^x`. For example, `5^-3=1/5^3`

 

Notice that these rules say that the base, `n`must be a “nonzero number”. When `n` is `0`, both `n^0` and `n` raised to a negative power are undefined.

Let’s take a look at how to simplify an algebraic expression with negative exponents in the numerator and the denominator. It’s helpful, especially at first, to separate the variables and simplify them individually:

Example

 

Problem

Simplify: `(b^-4a^-3)/c^-5`

 

 

`(b^-4/1)(a^-3/1)(1/c^-5)`

 

Rewrite as a product of fractions

 

`(1/b^4)(1/a^3)(1/(1/c^5))`

 

Rewrite variables with negative powers following the rule for negative exponents: `a^-n=1/a^n`

 

`(1/b^4)(1/a^3)(c^5/1)`

 

Simplify division by a fraction

 

 

`(c^5)/(b^4a^3)`

 

Multiply fractions

 

Answer

`c^5/(b^4a^3)`

 

Products and Quotients of Powers

Now that we’ve seen how to simplify variables with exponents, let’s learn how to multiply and divide them.

We’ll start with finding the product of two exponential terms with the same baseThe value that is raised to a power when a number is written in exponential notation. In the term `5^3`, `5` is the base and `3` is the exponent. . To find the product of powersMultiplication of two or more values in exponential form that have the same base—the base stays the same and the exponents are added. with the same base, just add the exponents and keep the base the same. Consider the example `x^2*x^3`. We could rewrite this problem as follows:

`x^2*x^3=(x*x)(x*x*x)=x*x*x*x*x=x^5`    

Or we could just add the exponents:

`x^2*x^3=x^(2+3)=x^5`

That second method is a lot easier.

Division with exponential notation has a similar shortcut. To find the quotient of powersDivision of two or more values in exponential form that have the same base—the base stays the same and the exponent in the denominator is subtracted from the exponent in the numerator. that have the same base, subtract the exponents. Here’s how it works:

Example

 

Problem

Simplify: `(9x^7)/x^3`

 

 

`(9*x*x*x*x*x*x*x)/(x*x*x`

 

Rewrite exponential notation

`(9/1)(x/1)(x/1)(x/1)(x/1)(x/1)(x/1)(x/1)(1/x)(1/x)(1/x)`

 

Rewrite as product of fractions

`(9/1)[x/1*1/x][x/1*1/x][x/1*1/x](x/1)(x/1)(x/1)(x/1)`

 

Use associative property to regroup factors

`(9/1)(x/1)(x/1)(x/1)(x/1)`

 

Because `[x/1*1/x]=x/x=1`

 

`9x^4`

Product of powers

Answer

`9x^4`

 

 

Here are a few additional examples:

`(9x^7)/x^3=9x^4`

`(-x^5)/x^2=-x^3`

`(15x^2y^8)/(3xy^6)=5xy^2`

`(4x^2y^3)/(8x^9y^2)=y/(2x^7)`

 

Rules for Multiplying and Dividing with Exponents

When multiplying two terms with the same base, add the exponents: `x^c*x^d=x^(c+d)`

When dividing two terms with the same base, subtract the exponent in the denominator from the exponent in the numerator: `b^a/b^c=b^(a-c)`

Note: `0^0` is undefined, so these rules do not apply in that situation.

 

Simplify the expression: `(30l^2m^3n^5)/(5lm^2n)` 

 

A) `25lmn^4` 

 

B) `6l^3m^5n^6`

 

C) `6l^2m^(3/2)n^5` 

 

D) `6lmn^4` 

 

Power of a Power

Exponential notation is a shorter way of writing repeated multiplication. When a variable expression with exponents is raised to a power, we could apply the rules for multiplying powers:

Example

Problem

Simplify: `(xyz)^3` 

 

 

 

`(xyz)(xyz)(xyz)`

Write as factors

`(x*x*x)(y*y*y)(z*z*z)`

Regroup

 

`x^(1+1+1)y^(1+1+1)z^(1+1+1)`

Add exponents with the same base

Answer

`x^3y^3z^3`

 

 

That worked, but it was cumbersome. There’s an easier way. Notice that the final exponent of each variable was the product of the exponent inside the parentheses, `1`, and the exponent outside the parentheses, `3`. To find the power of a powerRaising a value written in exponential notation to a power as in `(x^2)^3`. , multiply the exponents.

Let’s try it again:

Example

Problem

Simplify: `(3c^2d)^4` 

 

 

`(3^(1*4))(c^(2*4))(d^(1*4))`

Multiply the exponents

 

 

`3^4c^8d^4`

Simplify

 

 

`81c^8d^4`

Simplify

 

Answer

`81c^8d^4`

 

 

That was quicker!

Power of a Power:

To raise a power to a power, multiply the exponents. `(n^x)^y=n^(xy)`

 

Simplify: `(3a^2b^4c^2)^3` 

 

A) `9a^6b^12c^6` 

 

B) `27a^6b^12c^6` 

 

C) `27a^5b^7c^5` 

 

D) `33a^23b^43c^23` 

 

Summary

The rules of exponents apply to both numbers and variables. In brief, these rules are as follows:

Exponent of `0` or `1` and Negative Exponents

Any number or variable raised to a power of `1` is the number itself. `n^1=n` 

Any non-zero number or variable raised to a power of `0` is equal to `1``n^0=1` 

Any non-zero number `n` and any integer `x``n^-x=1/n^x`. For example, `5^-3=1/5^3`.  

Product of a Power

When multiplying two terms with the same base, add the exponents. `x^c*x^d=x^(c+d)` 

Quotient of a Power

When dividing two terms with the same base, subtract the exponent in the denominator from the exponent in the numerator. `b^a/b^c=b^(a-c)` 

Power of a Power:

To raise a power to a power, multiply the exponents. `(n^x)^y=n^(xy)` 

The rules of exponents provide accurate and efficient shortcuts for simplifying variables in exponential notation.