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Which of the following has the same value as one and a quarter to the power of negative two divided by one and a quarter to the power of negative three?
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Is it (A) four-fifths, (B) five-quarters, (C) five-quarters to the power of three over two, (D) five-quarters to the power of negative five, or (E) the square root of one and a quarter to the power of negative three?
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In order to answer this question, we need to recall one of our laws of exponents or indices.
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𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏.
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When dividing two terms with the same base, we can subtract the exponents.
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In this question, we need to subtract negative three from negative two.
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This is the same as adding three to negative two.
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Negative two plus three is equal to one.
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So, our expression simplifies to one and a quarter to the power of one.
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Any number raised to the power of one is itself.
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So, our answer is one and a quarter.
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This does not match any of our options, so we need to turn our mixed number into a top-heavy or improper fraction.
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We know that in one whole one, we have four-quarters.
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This means that one and a quarter is the same as five-quarters or five over four.
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A quick way to calculate the numerator when converting a mixed number is to multiply the whole number by the denominator and then adding the numerator.
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One multiplied by four plus one is equal to five.
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Our denominator when converting stays the same.
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The correct answer is option (B).
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One and a quarter to the power of negative two divided by one and a quarter to the power of negative three is five over four or five-quarters.