Product Rule of Exponents Worksheets (printable, online, answers ... - Free Printable

To solve the problems in the worksheet, we will use the Product Rule of Exponents, which states:

\[
a^m \times a^n = a^{m+n}
\]

This rule applies when multiplying two powers with the same base. Let's solve each problem step by step.

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Left Column

1. Simplify \( 3^2 \times 3^6 \)

- Using the Product Rule: \( a^m \times a^n = a^{m+n} \)
- Here, \( a = 3 \), \( m = 2 \), and \( n = 6 \)
- So, \( 3^2 \times 3^6 = 3^{2+6} = 3^8 \)

Answer: \( 3^8 \)

2. Simplify \( 4^2 \times 4^9 \)

- Using the Product Rule: \( a^m \times a^n = a^{m+n} \)
- Here, \( a = 4 \), \( m = 2 \), and \( n = 9 \)
- So, \( 4^2 \times 4^9 = 4^{2+9} = 4^{11} \)

Answer: \( 4^{11} \)

3. Simplify \( 8^2 \times 8^0 \)

- Using the Product Rule: \( a^m \times a^n = a^{m+n} \)
- Here, \( a = 8 \), \( m = 2 \), and \( n = 0 \)
- Recall that any number raised to the power of 0 is 1: \( 8^0 = 1 \)
- So, \( 8^2 \times 8^0 = 8^{2+0} = 8^2 \)

Answer: \( 8^2 \)

4. Simplify \( 7^3 \times 7^{11} \)

- Using the Product Rule: \( a^m \times a^n = a^{m+n} \)
- Here, \( a = 7 \), \( m = 3 \), and \( n = 11 \)
- So, \( 7^3 \times 7^{11} = 7^{3+11} = 7^{14} \)

Answer: \( 7^{14} \)

5. Simplify \( 6^5 \times 6 \)

- Note that \( 6 \) can be written as \( 6^1 \)
- Using the Product Rule: \( a^m \times a^n = a^{m+n} \)
- Here, \( a = 6 \), \( m = 5 \), and \( n = 1 \)
- So, \( 6^5 \times 6 = 6^{5+1} = 6^6 \)

Answer: \( 6^6 \)

6. Simplify \( 2 \times 2^3 \times 2^6 \)

- First, rewrite \( 2 \) as \( 2^1 \)
- Using the Product Rule repeatedly: \( a^m \times a^n \times a^p = a^{m+n+p} \)
- Here, \( a = 2 \), \( m = 1 \), \( n = 3 \), and \( p = 6 \)
- So, \( 2 \times 2^3 \times 2^6 = 2^{1+3+6} = 2^{10} \)

Answer: \( 2^{10} \)

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Right Column

1. Simplify \( x^9 \times x^4 \)

- Using the Product Rule: \( a^m \times a^n = a^{m+n} \)
- Here, \( a = x \), \( m = 9 \), and \( n = 4 \)
- So, \( x^9 \times x^4 = x^{9+4} = x^{13} \)

Answer: \( x^{13} \)

2. Simplify \( y^2 \times y^7 \)

- Using the Product Rule: \( a^m \times a^n = a^{m+n} \)
- Here, \( a = y \), \( m = 2 \), and \( n = 7 \)
- So, \( y^2 \times y^7 = y^{2+7} = y^9 \)

Answer: \( y^9 \)

3. Simplify \( b \times b^5 \)

- Note that \( b \) can be written as \( b^1 \)
- Using the Product Rule: \( a^m \times a^n = a^{m+n} \)
- Here, \( a = b \), \( m = 1 \), and \( n = 5 \)
- So, \( b \times b^5 = b^{1+5} = b^6 \)

Answer: \( b^6 \)

4. Simplify \( x^3 \times x^4 \)

- Using the Product Rule: \( a^m \times a^n = a^{m+n} \)
- Here, \( a = x \), \( m = 3 \), and \( n = 4 \)
- So, \( x^3 \times x^4 = x^{3+4} = x^7 \)

Answer: \( x^7 \)

5. Simplify \( w^4 \times w \)

- Note that \( w \) can be written as \( w^1 \)
- Using the Product Rule: \( a^m \times a^n = a^{m+n} \)
- Here, \( a = w \), \( m = 4 \), and \( n = 1 \)
- So, \( w^4 \times w = w^{4+1} = w^5 \)

Answer: \( w^5 \)

6. Simplify \( x^2 \times x^3 \times x^4 \)

- Using the Product Rule repeatedly: \( a^m \times a^n \times a^p = a^{m+n+p} \)
- Here, \( a = x \), \( m = 2 \), \( n = 3 \), and \( p = 4 \)
- So, \( x^2 \times x^3 \times x^4 = x^{2+3+4} = x^9 \)

Answer: \( x^9 \)

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Final Answers

\[
\boxed{
\begin{array}{ll}
3^2 \times 3^6 = 3^8 & x^9 \times x^4 = x^{13} \\
4^2 \times 4^9 = 4^{11} & y^2 \times y^7 = y^9 \\
8^2 \times 8^0 = 8^2 & b \times b^5 = b^6 \\
7^3 \times 7^{11} = 7^{14} & x^3 \times x^4 = x^7 \\
6^5 \times 6 = 6^6 & w^4 \times w = w^5 \\
2 \times 2^3 \times 2^6 = 2^{10} & x^2 \times x^3 \times x^4 = x^9 \\
\end{array}
}
\]