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Which of the following has the same value as three-quarters squared divided by three-quarters to the power of negative three?
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Is it (A) three-quarters to the power of negative two-thirds, (B) three-quarters to the power of negative six, (C) three-quarters, (D) four-thirds, or (E) three to the fifth power over four to the fifth power?
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In this question, we’re dividing two equal factions with integer powers.
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And as long as these fractions are not mixed numbers — that is, they’re simply one integer over another — we can use the standard index laws to evaluate this.
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The first law we’re interested in says that 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is 𝑥 to the power of 𝑎 minus 𝑏.
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That is to say, when we divide two numbers whose base is equal, we simply subtract their exponents.
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And so three-quarters squared divided by three-quarters to the power of negative three is three-quarters to the power of two minus negative three.
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Two minus negative three is two plus three, which is five.
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So we have three-quarters to the fifth power.
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But another one of our laws of indices says that we can distribute the power or the exponent over the numerator and denominator of our fraction.
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So three-quarters to the fifth power is the same as three to the fifth power over four to the fifth power.
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And we don’t actually need to evaluate this since this is one of our options; it’s (E).
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Three-quarters squared divided by three-quarters to the power of negative three is the same as three to the fifth power over four to the fifth power.