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Algebra Expressions with Exponents

Calculate values of numbers with exponents

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Algebra Expressions with Exponents
License: CC BY-NC 3.0

Sam is preparing to have an epic water balloon fight with his best friend Josh. He is at home making his water balloons and wonders what the total volume of water is in his 50 balloons. He knows his balloons are approximately spheres. He also learned that the volume of a sphere is approximately \begin{align*}4.19r^3\end{align*} where \begin{align*}r\end{align*} is the radius of the sphere. How could Sam determine the total volume of water in his balloons if the radius of each balloon is approximately 2 inches?

In this concept, you will learn how to simplify exponential expressions using exponent rules.

Simplifying Algebraic Expressions with Exponents

Sometimes you need to multiply a number or a variable by itself many times.

Here is an example.

\begin{align*}x \cdot x \cdot x \cdot x \cdot x\end{align*} is \begin{align*}x\end{align*} multiplied by itself 5 times.

To avoid having to write out the \begin{align*}x\end{align*} again and again, you can use an exponent. Whole number exponents are shorthand for repeated multiplication of a number or a variable by itself. 

\begin{align*}x \cdot x \cdot x \cdot x \cdot x=x^5\end{align*}

In this example, 5 is the exponent and \begin{align*}x\end{align*} is the base. The exponent indicates how many times the base is being multiplied by itself.

When you use an exponent to write an expression you are using exponential form. \begin{align*}x^5\end{align*} is exponential form. When you write out the expression using multiplication without an exponent you are using expanded form. \begin{align*}x \cdot x \cdot x \cdot x \cdot x\end{align*} is expanded form.

Here is an example.

Write the following in expanded form: \begin{align*}y^{12}\end{align*}

First, notice that in this expression \begin{align*}y\end{align*} is the base and 12 is the exponent. \begin{align*}y^{12}\end{align*} means \begin{align*}y\end{align*} is being multiplied by itself 12 times.

Next, write the expression in expanded form without an exponent.

\begin{align*}y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y\end{align*}

It's much easier to write the expression in exponential form!

The answer is \begin{align*}y^{12}=y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y\end{align*}.

When you have a variable expression with more than one exponent, you can often simplify.

Here is an example.

Simplify \begin{align*}m^3 \cdot m^2\end{align*}.

First, write the expression in expanded form by expanding each part of the expression.

\begin{align*}m^3 \cdot m^2=m \cdot m \cdot m \cdot m \cdot m\end{align*}

Next, rewrite your expression in exponential form using just one base and one exponent. 

\begin{align*}m \cdot m \cdot m \cdot m \cdot m=m^5\end{align*}

The answer is \begin{align*}m^3 \cdot m^2=m^5\end{align*}.

Notice that the exponent in the answer is the sum of the original two exponents. This example helps to illustrate the first rule of exponents.

Rule 1: When multiplying two powers that have the same base, you can add the exponents. In symbols, \begin{align*}x^a \cdot x^b=x^{a+b}\end{align*}.

Let's apply this rule to the next example.

Simplify \begin{align*}(x^6 )(x^3 )\end{align*}.

First, remember that when parentheses are right next to one another it means multiplication.

\begin{align*}(x^6 )(x^3 )=x^6 \cdot x^3\end{align*}

Next, notice that the two bases are the same. This means you can apply Rule 1 and simplify by adding the exponents.

\begin{align*}x^6 \cdot x^3=x^{6+3}=x^9\end{align*}

If you are ever not sure, remember you can always write the expression in expanded form and then convert back to exponential form. 

\begin{align*}x^6 \cdot x^3=x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x=x^9\end{align*}

The answer is \begin{align*}(x^6 )(x^3 )=x^9\end{align*}.

You can also have an exponential term raised to a power.

Here is an example.

Simplify \begin{align*}(x^2 )^3\end{align*}.

First, focus on the exponent of 3 and start expanding the expression. \begin{align*}x^2\end{align*} is being multiplied by itself 3 times.

\begin{align*}(x^2 )^3=x^2 \cdot x^2 \cdot x^2\end{align*}

Now, you are multiplying three powers that each have the same base so you can use Rule 1 and add the exponents. 

\begin{align*}x^2 \cdot x^2 \cdot x^2=x^{2+2+2}=x^6\end{align*}

The answer is \begin{align*}(x^2 )^3=x^6\end{align*}.

Notice that the exponent in the answer is the product of the original two exponents. This example helps to illustrate the second rule of exponents.

Rule 2: When raising a power to a power, you can multiply the exponents. In symbols, \begin{align*}(x^a )^b=x^{ab}\end{align*}.

Let's apply this rule to the next example.

Simplify \begin{align*}(y^4 )^3\end{align*}.

First, notice that this expression is a power raised to another power. There are two exponents, but only one variable base. This means you can use Rule 2 and simplify by multiplying the exponents.

\begin{align*}(y^4 )^3=y^{4 \cdot 3}=y^{12}\end{align*}

Again, remember that you can always write your expression in expanded form and then convert back to exponential form if you want to check your answer.

\begin{align*}(y^4 )^3=y^4 \cdot y^4 \cdot y^4=y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y=y^{12}\end{align*}

The answer is \begin{align*}(y^4 )^3=y^{12}\end{align*}.

The final exponent rule that you need to know right now has to do with an exponent of 0.

Rule 3: Any nonzero number raised to the power of 0 is equal to 1. In symbols, \begin{align*}x^0=1\end{align*} as long as \begin{align*}x \neq 0\end{align*}.

Here is an example.

Simplify \begin{align*}15^0\end{align*}.

First, notice that the exponent is 0. This means you can use Rule 3.

\begin{align*}15^0=1\end{align*}

The answer is \begin{align*}15^0=1\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Sam and his upcoming epic water balloon fight.

He has 50 water balloons that each have a volume of approximately \begin{align*}4.19r^3\end{align*} where \begin{align*}r\end{align*} is the radius of each balloon. He measured his balloons and found that each has a radius of approximately 2 inches. Now he wonders what the total volume of water is in his balloons.

First, Sam can figure out the volume of water in one water balloon. He can substitute 2 in for the \begin{align*}r\end{align*} in his volume formula.

\begin{align*}4.19(2^3)\end{align*}

Next, he can expand the expression. 2 is being multiplied by itself 3 times. 

\begin{align*}4.19(2^3 )=4.19 \cdot 2 \cdot 2 \cdot 2\end{align*}

Then, he can multiply.

\begin{align*}4.19 \cdot 2 \cdot 2 \cdot 2=33.52\end{align*}

So each water balloon has 33.52 cubic inches of water.

Now, since he has 50 water balloons, to find the total volume of water in his balloons he can multiply the volume of water in one balloon by 50.

\begin{align*}50 \cdot 33.52=1676\end{align*}

The answer is that in total, Sam has 1676 cubic inches of water in his water balloons.

Example 2

Simplify \begin{align*}(x^6 )(x^2 )\end{align*}.

First, remember that when parentheses are right next to one another it means multiplication.

\begin{align*}(x^6 )(x^2 )=x^6 \cdot x^2\end{align*}

Next, notice that the two bases are the same. This means you can apply Rule 1 and simplify by adding the exponents. 

\begin{align*}x^6 \cdot x^2=x^{6+2}=x^8\end{align*}

The answer is \begin{align*}(x^6 )(x^2 )=x^8\end{align*}.

Example 3

Write the following in exponential form: \begin{align*}aaaaaaa\end{align*}

First, notice that \begin{align*}a\end{align*} is being multiplied by itself 7 times. \begin{align*}a\end{align*} will be your base and 7 will be your exponent.

The answer is \begin{align*}aaaaaaa=a^7\end{align*}.

Example 4

Simplify \begin{align*}(a^3 )(a^8 )\end{align*}.

First, remember that when parentheses are right next to one another it means multiplication.

\begin{align*}(a^3 )(a^8 )=a^3 \cdot a^8\end{align*}

Next, notice that the two bases are the same. This means you can apply Rule 1 and simplify by adding the exponents. 

\begin{align*}a^3 \cdot a^8=a^{3+8}=a^{11}\end{align*}

The answer is \begin{align*}(a^3 )(a^8 )=a^{11}\end{align*}.

Example 5

Simplify \begin{align*}(x^4 )^2\end{align*}.

First, notice that this expression is a power raised to another power. This means you can use Rule 2 and simplify by multiplying the exponents.

\begin{align*}(x^4 )^2=x^{4 \cdot 2}=x^8\end{align*}

The answer is \begin{align*}(x^4 )^2=x^8\end{align*}.

Review

Evaluate each expression.

  1. \begin{align*}2^3\end{align*} 
  2. \begin{align*}4^2\end{align*} 
  3. \begin{align*}5^2\end{align*} 
  4. \begin{align*}9^0\end{align*} 
  5. \begin{align*}5^3\end{align*} 
  6. \begin{align*}2^6\end{align*} 
  7. \begin{align*}3^3\end{align*} 
  8. \begin{align*}3^2+4^2\end{align*} 
  9. \begin{align*}5^3+2^2\end{align*} 
  10. \begin{align*}6^2 + 2^3\end{align*} 
  11. \begin{align*}6^2-5^2\end{align*} 
  12. \begin{align*}2^4-2^2\end{align*} 
  13. \begin{align*}7^2+3^3+2^2\end{align*} 

Simplify each expression.

  1. \begin{align*}(m^2 )(m^5 )\end{align*} 
  2. \begin{align*}(x^3 )(x^4 )\end{align*} 
  3. \begin{align*}(y^5)(y^3)\end{align*} 
  4. \begin{align*}(b^7)(b^2)\end{align*} 
  5. \begin{align*}(a^5)(a^2)\end{align*} 
  6. \begin{align*}(x^9)(x^3)\end{align*} 
  7. \begin{align*}(y^4)(y^5)\end{align*} 
  8. \begin{align*}(x^2)^4\end{align*} 
  9. \begin{align*}(y^5)^3\end{align*} 
  10. \begin{align*}(a^5)^4\end{align*} 
  11. \begin{align*}(x^2 )^8\end{align*} 
  12. \begin{align*}(b^3 )^4\end{align*} 

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.8.

Vocabulary

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.

Expanded Form

Expanded form refers to a base and an exponent written as repeated multiplication.

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.

Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

Parentheses

Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.

substitute

In algebra, to substitute means to replace a variable or term with a specific value.

Volume

Volume is the amount of space inside the bounds of a three-dimensional object.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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