Video: Evaluating Positive Rational Expressions with Positive Integer Exponents Raised to Positive Integer Exponents

Which of the following expressions has the same value as ((2/3)²)³? [A] 2⁶/3⁶ [B] (2/3)⁵ [C] 2⁸/3⁸ [D] (4/6)³ [E] (2/3)² × (2/3)²

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Video Transcript

Which of the following expressions has the same value as two-thirds squared all to the power of three? Is it A) two to the power of six divided by three to the power of six? Is it B) two-thirds to the power of five, C) two to the power of eight divided by three to the power of eight, D) four-sixths to the power of three, or E) two-thirds squared multiplied by two-thirds squared?

The first thing to consider in this problem is what happens to the exponents or indices. Using the laws of indices 𝑥 to the power of 𝑎 all to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 multiplied by 𝑏 tells us that we can multiply the exponents or indices.

In our example, two-thirds squared all to the power of three can be rewritten as two-thirds to the power of six as multiplying the powers two and three gives us six. Another one of our laws of indices or exponents tells us that any fraction 𝑎 over 𝑏 that is raised to the power of 𝑥 can be rewritten as 𝑎 to the power of 𝑥 divided by 𝑏 to the power of 𝑥. In our example, two-thirds to the power of six can be rewritten as two to the power of six divided by three to the power of six.

This means that the expression that has the same value as two-thirds squared all to the power of three is option A, two to the power of six divided by three to the power of six.

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