How to Solve Powers of Products and Quotients

How to Solve Powers of Products and Quotients

Learn how to take a product or a quotient to power by using powers of products and quotients rules.

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Step by step guide to solve powers of products and quotients

  • For any nonzero numbers \(a\) and \(b\) and any integer \(x\),
    \(\color{blue}{(ab)^x=a^x×b^x}\).

Powers of Products and Quotients – Example 1:

Simplify. \((6x^2 y^4)^2=\)

Solution:

Use Exponent’s rules: \(\color{blue}{(ab)^x=a^x×b^x}\)
\((6x^2y^4)^2=(6)^2(x^2)^2(y^4)^2=36x^{2×2}y^{4×2}=36x^4y^8\)

Powers of Products and Quotients – Example 2:

Simplify. \((\frac{5x}{2x^2})^2=\)

Solution:

First cancel the common factor: \(x→ (\frac{5x}{2x^2} )^2=(\frac{5}{2x})^2 \)
Use Exponent’s rules: \(\color{blue}{(\frac{a}{b})^c=\frac{a^c}{b^c}}\)
Then: \((\frac{5}{2x})^2=\frac{5^2}{(2x)^2} =\frac{25}{4x^2}\)

Powers of Products and Quotients – Example 3:

Simplify. \((3x^5 y^4)^2=\)

Solution:

Use Exponent’s rules: \(\color{ blue }{(x^a)^b=x^{a×b}}\)
\((3x^5 y^4)^2=(3)^2 (x^5)^2 (y^4)^2=9x^{5×2} y^{4×2}=9x^{10} y^8 \)

Powers of Products and Quotients – Example 4:

Simplify. \((\frac{2x}{3x^2} )^2=\)

Solution:

First cancel the common factor: \(x→ (\frac{2x}{3x^2} )^2=(\frac{2}{3x})^2 \)
Use Exponent’s rules: \(\color{blue}{(\frac{a}{b})^c=\frac{a^c}{b^c}}\)
Then: \((\frac{2}{3x})^2=\frac{2^2}{(3x)^2 }=\frac{4}{9x^2 }\)

Exercises for Solving Powers of Products and Quotients

Simplify.

  1. \(\color{blue}{(2x^6y^8)^2}\)
  2. \(\color{blue}{(2x^3x)^3}\)
  3. \(\color{blue}{(2x^9 x^6)^3}\)
  4. \(\color{blue}{(5x^{10}y^3)^3}\)
  5. \(\color{blue}{(4x^3 x^2)^2}\)
  6. \(\color{blue}{(3x^3 5x)^2}\)

Download Powers of Products and Quotients Worksheet

  1. \(\color{blue}{4x^{12}y^{16}}\)
  2. \(\color{blue}{8x^{64}}\)
  3. \(\color{blue}{8x^{45} }\)
  4. \(\color{blue}{125x^{30}y^9}\)
  5. \(\color{blue}{16x^{10}}\)
  6. \(\color{blue}{225x^8}\)

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