Question Video: Evaluating Negative Single-Term Rational Expressions with Negative Integer Exponents | Nagwa Question Video: Evaluating Negative Single-Term Rational Expressions with Negative Integer Exponents | Nagwa

Question Video: Evaluating Negative Single-Term Rational Expressions with Negative Integer Exponents Mathematics • First Year of Preparatory School

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Which of the following is equal to (−(3/4))⁻²? [A] 16/9 [B] −(16/9) [C] −(−6/−8) [D] 9/16 [E] −(9/16)

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Video Transcript

Which of the following is equal to negative three-quarters all raised to the power of negative two? Option (A) 16 over nine. Option (B) negative 16 over nine. Option (C) negative one times negative six over negative eight. Option (D) nine over 16. Or is it option (E) negative nine over 16?

In this question, we are given an expression involving a fraction raised to a negative exponent. And we are asked to determine which of five given expressions is equal to the given expression. To do this, we can start by noting that the exponent of the expression is negative. We can evaluate expressions with negative exponents by recalling that a negative exponent is the same as the positive exponent where we also take the reciprocal of the base. So, 𝑎 over 𝑏 all raised to the power of negative 𝑛 is equal to 𝑏 over 𝑎 all raised to the power of 𝑛. We can apply this to the given expression.

First, we can rewrite the base of the expression to get negative three over four all raised to the power of negative two. Next, we take the reciprocal of the base so that we can change the power to be positive two. This is the same as setting 𝑛 equals two, 𝑎 equals negative three, and 𝑏 equals four in our exponent rule. We obtain four over negative three all squared. We can simplify this expression further by recalling that raising a fraction to an exponent is equivalent to raising its numerator and denominator separately to that exponent.

So, 𝑏 over 𝑎 all raised to the power of 𝑛 is equal to 𝑏 raised to the 𝑛th power over 𝑎 raised to the 𝑛th power. We can use this to rewrite our expression. We want to square the numerator and denominator separately, so we have four squared over negative three squared. We can now evaluate this expression by recalling that squaring a number means multiplying it by itself. So, 𝑎 squared is 𝑎 times 𝑎.

Applying this result to the numerator and denominator gives us four times four over negative three times negative three. We can calculate that four times four is 16 and negative three times negative three is nine, since a negative times a negative is a positive. This gives us 16 over nine, which we can see matches option (A).

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