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Which of the following expressions has the same value as two-thirds squared all to the power of three?
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Is it A) two to the power of six divided by three to the power of six?
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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?
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The first thing to consider in this problem is what happens to the exponents or indices.
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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.
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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.
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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.
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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.
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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.