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Question Video: Simplifying an Algebraic Expression Involving Negative and Fractional Exponents Mathematics • 9th Grade

True or false: The simplified form of ๐‘ฅ^(2/5)/๐‘ฆ^(โˆ’1/3) is ๐‘ฅ^(2/5) ๐‘ฆ^(1/3).

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Video Transcript

True or false: The simplified form of ๐‘ฅ to the power of two-fifths divided by ๐‘ฆ to the power of negative one-third is ๐‘ฅ to the power of two-fifths multiplied by ๐‘ฆ to the power of one-third.

In order to answer this question, we will recall one of our rules of exponents which helps us deal with negative exponents or powers. This states that ๐‘Ž to the power of negative ๐‘› is equal to one over ๐‘Ž to the power of ๐‘›. A negative power is the same as taking the reciprocal. Rearranging this equation, we have ๐‘Ž to the power of ๐‘› is equal to one over ๐‘Ž to the power of negative ๐‘›.

Letโ€™s now consider how we can use this in the given example. We are trying to divide ๐‘ฅ to the power of two-fifths by ๐‘ฆ to the power of negative one-third. This can be rewritten as ๐‘ฅ to the power of two-fifths divided by one over ๐‘ฆ to the power of one third. We recall that dividing by a fraction is the same as multiplying by the reciprocal of this fraction.

One way of remembering this is using the acronym KCF, which stands for Keep, Change, Flip. We keep the first term the same, change the division to a multiplication, and flip the second term. This gives us ๐‘ฅ to the power of two-fifths multiplied by ๐‘ฆ to the power of one-third, which can be rewritten without the multiplication sign between both terms.

The simplified form of ๐‘ฅ to the power of two-fifths divided by ๐‘ฆ to the power of negative one-third is equal to ๐‘ฅ to the power of two-fifths multiplied by ๐‘ฆ to the power of one-third. Therefore, the statement is true.

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