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

# Question Video: Evaluating Positive 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 (2/3)⁻³? [A] 27/8 [B] 8/27 [C] −8/27 [D] −6/−9 [E] −27/8

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### Video Transcript

Which of the following is equal to two-thirds to the power of negative three? The options are (A) 27 over eight, (B) eight over 27, (C) negative eight over 27, (D) negative six over negative nine, or (E) negative 27 over eight.

So, to solve this problem, what we’re gonna do is we’re going to use an adaptation of the most common exponent rule. And that exponent rule is 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. Because, in fact, what this really means is 𝑥 to the power of negative 𝑛 is equal to one to the power of 𝑛 over 𝑥 to the power of 𝑛. However, as one to the power of 𝑛 is just one, we write one over 𝑥 to the power of 𝑛. So therefore, if we’ve got 𝑥 over 𝑦 to the power of negative 𝑛, this is gonna be equal to 𝑦 to the power of 𝑛 over 𝑥 to the power of 𝑛. Cause what we do is we find the reciprocal of our fraction and we put each of the terms raised to the power of 𝑛.

So therefore, if we got two-thirds, or two over three, to the power of negative three, well, first of all, what we do is we find the reciprocal. And the reciprocal is what we get if we flip the numerator and denominator. So, two-thirds becomes three-halves or three over two. Well, then, what we’re gonna do is raise both the numerator and the denominator to the power of three because, in our example, this is our 𝑛. Well, three cubed means three multiplied by three multiplied by three, and two cubed means two multiplied by two multiplied by two.

So therefore, we can say that two-thirds to the power of negative three is gonna be equal to 27 over eight. And that’s because three times three times three is 27 and two times two times two is eight. So therefore, we can say that the correct answer from our list is going to be answer (A) because this is 27 over eight or twenty-seven eighths.

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