Multiplication Property of Exponents worksheets - Free Printable

Let's solve each problem step by step using the Multiplication Property of Exponents. This property includes:

1. Product of Powers Rule: $ a^m \times a^n = a^{m+n} $
2. Power of a Power Rule: $ (a^m)^n = a^{m \times n} $
3. Power of a Product Rule: $ (ab)^n = a^n b^n $
4. Coefficient multiplication: Multiply coefficients separately.

We’ll go through each problem and simplify accordingly.

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1) $ (6x^2)^2 $

Use power of a product rule:
$$
(6x^2)^2 = 6^2 \cdot (x^2)^2 = 36 \cdot x^{4} = \boxed{36x^4}
$$

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2) $ 4^3 \times 4^7 $

Use product of powers:
$$
4^{3+7} = 4^{10} = \boxed{4^{10}}
$$

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3) $ 10^5 \times 10 $

$ 10 = 10^1 $, so:
$$
10^5 \times 10^1 = 10^{5+1} = \boxed{10^6}
$$

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4) $ (11a^6)^2 $

Use power of a product:
$$
11^2 \cdot (a^6)^2 = 121 \cdot a^{12} = \boxed{121a^{12}}
$$

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5) $ 7x^6 \times x^3 \times x $

Combine exponents: $ x^6 \cdot x^3 \cdot x^1 = x^{6+3+1} = x^{10} $, and coefficient is 7:
$$
\boxed{7x^{10}}
$$

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6) $ 2x^2 \times 3yx^6 $

Multiply coefficients: $ 2 \times 3 = 6 $, variables: $ x^2 \cdot x^6 = x^8 $, and $ y $ stays.
$$
\boxed{6yx^8}
$$

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7) $ 6x^2 \times x^7 \times x $

$ x^2 \cdot x^7 \cdot x^1 = x^{10} $, coefficient 6:
$$
\boxed{6x^{10}}
$$

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8) $ 7y^2x^3 \times 3y^3x^3 $

Coefficients: $ 7 \times 3 = 21 $
$ y^2 \cdot y^3 = y^5 $, $ x^3 \cdot x^3 = x^6 $
$$
\boxed{21y^5x^6}
$$

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9) $ 2x^7 \times x^5 \times x $

$ x^7 \cdot x^5 \cdot x = x^{13} $, coefficient 2:
$$
\boxed{2x^{13}}
$$

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10) $ 5y^4x^2 \times 3y^3x^4 $

Coefficients: $ 5 \times 3 = 15 $
$ y^4 \cdot y^3 = y^7 $, $ x^2 \cdot x^4 = x^6 $
$$
\boxed{15y^7x^6}
$$

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11) $ 7^4 \times 7 $

$ 7^4 \times 7^1 = 7^{5} = \boxed{7^5} $

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12) $ 3^7 \times 3 $

$ 3^7 \times 3^1 = 3^{8} = \boxed{3^8} $

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13) $ 10x^4 \times 3yx^7 $

Coefficients: $ 10 \times 3 = 30 $, $ x^4 \cdot x^7 = x^{11} $, $ y $ stays
$$
\boxed{30yx^{11}}
$$

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14) $ 8x^5 \times 2yx^4 $

$ 8 \times 2 = 16 $, $ x^5 \cdot x^4 = x^9 $, $ y $ stays
$$
\boxed{16yx^9}
$$

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15) $ 4^3 \times 4 $

$ 4^3 \times 4^1 = 4^{4} = \boxed{4^4} $

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16) $ (3x^3)^2 $

$ 3^2 \cdot (x^3)^2 = 9 \cdot x^6 = \boxed{9x^6} $

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17) $ 10y^5x^2 \times 3y^3x^3 $

Coefficients: $ 10 \times 3 = 30 $
$ y^5 \cdot y^3 = y^8 $, $ x^2 \cdot x^3 = x^5 $
$$
\boxed{30y^8x^5}
$$

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18) $ 3^6 \times 3^2 $

$ 3^{6+2} = 3^8 = \boxed{3^8} $

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19) $ 8^3 \times 8 $

$ 8^3 \times 8^1 = 8^4 = \boxed{8^4} $

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20) $ 2y^5x^3 \times 3y^2x^4 $

Coefficients: $ 2 \times 3 = 6 $
$ y^5 \cdot y^2 = y^7 $, $ x^3 \cdot x^4 = x^7 $
$$
\boxed{6y^7x^7}
$$

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21) $ 9x^4 \times 3yx^5 $

$ 9 \times 3 = 27 $, $ x^4 \cdot x^5 = x^9 $, $ y $ stays
$$
\boxed{27yx^9}
$$

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22) $ 11x^6 \times 4yx^2 $

$ 11 \times 4 = 44 $, $ x^6 \cdot x^2 = x^8 $, $ y $ stays
$$
\boxed{44yx^8}
$$

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23) $ 5^7 \times 5^5 $

$ 5^{7+5} = 5^{12} = \boxed{5^{12}} $

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24) $ 8^4 \times 8^6 $

$ 8^{4+6} = 8^{10} = \boxed{8^{10}} $

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25) $ 4y^7x^4 \times 3y^2x^4 $

Coefficients: $ 4 \times 3 = 12 $
$ y^7 \cdot y^2 = y^9 $, $ x^4 \cdot x^4 = x^8 $
$$
\boxed{12y^9x^8}
$$

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26) $ 9y^2x^4 \times 4y^2x^4 $

$ 9 \times 4 = 36 $, $ y^2 \cdot y^2 = y^4 $, $ x^4 \cdot x^4 = x^8 $
$$
\boxed{36y^4x^8}
$$

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27) $ 5x^4 \times x^6 \times x $

$ x^4 \cdot x^6 \cdot x = x^{11} $, coefficient 5
$$
\boxed{5x^{11}}
$$

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28) $ 3x^2 \times x^3 \times x $

$ x^2 \cdot x^3 \cdot x = x^{6} $, coefficient 3
$$
\boxed{3x^6}
$$

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29) $ 6^8 \times 6 $

$ 6^8 \times 6^1 = 6^9 = \boxed{6^9} $

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30) $ 8y^7x^2 \times 3y^4x^3 $

Coefficients: $ 8 \times 3 = 24 $
$ y^7 \cdot y^4 = y^{11} $, $ x^2 \cdot x^3 = x^5 $
$$
\boxed{24y^{11}x^5}
$$

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✔ Final Answers:

1) $ 36x^4 $
2) $ 4^{10} $
3) $ 10^6 $
4) $ 121a^{12} $
5) $ 7x^{10} $
6) $ 6yx^8 $
7) $ 6x^{10} $
8) $ 21y^5x^6 $
9) $ 2x^{13} $
10) $ 15y^7x^6 $
11) $ 7^5 $
12) $ 3^8 $
13) $ 30yx^{11} $
14) $ 16yx^9 $
15) $ 4^4 $
16) $ 9x^6 $
17) $ 30y^8x^5 $
18) $ 3^8 $
19) $ 8^4 $
20) $ 6y^7x^7 $
21) $ 27yx^9 $
22) $ 44yx^8 $
23) $ 5^{12} $
24) $ 8^{10} $
25) $ 12y^9x^8 $
26) $ 36y^4x^8 $
27) $ 5x^{11} $
28) $ 3x^6 $
29) $ 6^9 $
30) $ 24y^{11}x^5 $

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Let me know if you'd like these in a printable format or with explanations for any specific ones!