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Which of the following is equal to two and a half cubed multiplied by two and a half squared?
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Is it (A) 3125 over 32, (B) 15625 over 64, (C) 75 over two, (D) two and one over 32, or (E) two and a half squared all to the power of five?
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As the fractions inside the brackets are equal, one way to start this problem would be to use one of the laws of exponents or indices.
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This states that 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏.
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When multiplying a number raised to two exponents, we can add the exponents or indices.
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In this question, two and a half cubed multiplied by two and a half squared is equal to two and a half to the power of five.
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This is because three plus two is equal to five.
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Our next step is to convert the mixed number two and a half into an improper or top-heavy fraction.
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Two is the same as four-halves or four over two.
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This means that two and a half is equal to four-halves plus one-half.
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This is equal to five-halves.
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So two and a half can be rewritten as five over two.
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A quicker way to work out the numerator of this improper fraction is to multiply the whole number, two, by the denominator, two, and then add the numerator, one.
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Two multiplied by two is equal to four.
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Adding one gives us five.
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Going back to our question, two and a half to the power of five can be rewritten as five over two to the power of five.
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Another one of our laws of indices or exponents states 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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This means that five over two to the power of five can be rewritten as five to the power of five divided by two to the power of five.
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Five to the power of five is equal to 3125 as five multiplied by five multiplied by five multiplied by five multiplied by five is equal to 3125.
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In the same way, two to the power of five is equal to 32.
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We can therefore say that option A, 3125 over 32, is equal to two and a half cubed multiplied by two and a half squared.