Properties of Exponents Worksheet: (4 A / 2 B) 1/x y | Download ... - Free Printable

The worksheet provided involves simplifying expressions involving exponents. Below, I will solve a few of the problems step by step to illustrate the process. If you need solutions for specific problems, please let me know!

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Problem 7: Simplify \( x^3 \cdot x^4 \)

Solution:
When multiplying powers with the same base, we add the exponents:
\[
x^3 \cdot x^4 = x^{3+4} = x^7
\]

Answer:
\[
\boxed{x^7}
\]

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Problem 8: Simplify \( x^5 \div x^2 \)

Solution:
When dividing powers with the same base, we subtract the exponents:
\[
x^5 \div x^2 = x^{5-2} = x^3
\]

Answer:
\[
\boxed{x^3}
\]

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Problem 9: Simplify \( (4a^3b^2)^3 \)

Solution:
When raising a product to a power, we raise each factor to that power:
\[
(4a^3b^2)^3 = 4^3 \cdot (a^3)^3 \cdot (b^2)^3
\]
Calculate each part:
\[
4^3 = 64, \quad (a^3)^3 = a^{3 \cdot 3} = a^9, \quad (b^2)^3 = b^{2 \cdot 3} = b^6
\]
Combine the results:
\[
(4a^3b^2)^3 = 64a^9b^6
\]

Answer:
\[
\boxed{64a^9b^6}
\]

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Problem 10: Simplify \( \frac{x^6}{x^3} \)

Solution:
When dividing powers with the same base, we subtract the exponents:
\[
\frac{x^6}{x^3} = x^{6-3} = x^3
\]

Answer:
\[
\boxed{x^3}
\]

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Problem 11: Simplify \( x^3 \cdot x^{-2} \)

Solution:
When multiplying powers with the same base, we add the exponents:
\[
x^3 \cdot x^{-2} = x^{3 + (-2)} = x^{3-2} = x^1 = x
\]

Answer:
\[
\boxed{x}
\]

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Problem 12: Simplify \( x^0 \cdot x^2 \cdot x^{-1} \)

Solution:
First, recall that \( x^0 = 1 \). Then, multiply the powers:
\[
x^0 \cdot x^2 \cdot x^{-1} = 1 \cdot x^{2 + (-1)} = x^{2-1} = x^1 = x
\]

Answer:
\[
\boxed{x}
\]

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Problem 13: Simplify \( \left( \frac{2a^3}{3b^2} \right)^2 \)

Solution:
When raising a fraction to a power, we raise both the numerator and the denominator to that power:
\[
\left( \frac{2a^3}{3b^2} \right)^2 = \frac{(2a^3)^2}{(3b^2)^2}
\]
Simplify the numerator and the denominator:
\[
(2a^3)^2 = 2^2 \cdot (a^3)^2 = 4 \cdot a^{3 \cdot 2} = 4a^6
\]
\[
(3b^2)^2 = 3^2 \cdot (b^2)^2 = 9 \cdot b^{2 \cdot 2} = 9b^4
\]
Combine the results:
\[
\left( \frac{2a^3}{3b^2} \right)^2 = \frac{4a^6}{9b^4}
\]

Answer:
\[
\boxed{\frac{4a^6}{9b^4}}
\]

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Problem 14: Simplify \( (2x^2y^3)^2 \cdot (3xy^2)^3 \)

Solution:
First, simplify each term separately:
\[
(2x^2y^3)^2 = 2^2 \cdot (x^2)^2 \cdot (y^3)^2 = 4 \cdot x^{2 \cdot 2} \cdot y^{3 \cdot 2} = 4x^4y^6
\]
\[
(3xy^2)^3 = 3^3 \cdot (x)^3 \cdot (y^2)^3 = 27 \cdot x^3 \cdot y^{2 \cdot 3} = 27x^3y^6
\]
Now, multiply the results:
\[
(2x^2y^3)^2 \cdot (3xy^2)^3 = 4x^4y^6 \cdot 27x^3y^6
\]
Combine like terms:
\[
4 \cdot 27 \cdot x^{4+3} \cdot y^{6+6} = 108x^7y^{12}
\]

Answer:
\[
\boxed{108x^7y^{12}}
\]

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Problem 15: Simplify \( \frac{(2x^3y^2)^2}{(4xy^3)^2} \)

Solution:
Simplify the numerator and the denominator separately:
\[
(2x^3y^2)^2 = 2^2 \cdot (x^3)^2 \cdot (y^2)^2 = 4 \cdot x^{3 \cdot 2} \cdot y^{2 \cdot 2} = 4x^6y^4
\]
\[
(4xy^3)^2 = 4^2 \cdot (x)^2 \cdot (y^3)^2 = 16 \cdot x^2 \cdot y^{3 \cdot 2} = 16x^2y^6
\]
Now, divide the results:
\[
\frac{(2x^3y^2)^2}{(4xy^3)^2} = \frac{4x^6y^4}{16x^2y^6}
\]
Simplify the coefficients and the variables:
\[
\frac{4}{16} = \frac{1}{4}, \quad x^6 \div x^2 = x^{6-2} = x^4, \quad y^4 \div y^6 = y^{4-6} = y^{-2}
\]
Combine the results:
\[
\frac{4x^6y^4}{16x^2y^6} = \frac{1}{4} \cdot x^4 \cdot y^{-2} = \frac{x^4}{4y^2}
\]

Answer:
\[
\boxed{\frac{x^4}{4y^2}}
\]

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If you need solutions for more problems, feel free to ask!