Video: Dividing Positive Rational Expressions by Mixed Positive and Negative Integer Exponents

Which of the following has the same value as (3/4)² ÷ (3/4)⁻³? [A] (3/4)^(−2/3) [B] (3/4)⁻⁶ [C] 3/4 [D] 4/3 [E] 3⁵/4⁵

01:25

Video Transcript

Which of the following has the same value as three-quarters squared divided by three-quarters to the power of negative three? 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?

In this question, we’re dividing two equal factions with integer powers. 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. 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. That is to say, when we divide two numbers whose base is equal, we simply subtract their exponents.

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. Two minus negative three is two plus three, which is five. So we have three-quarters to the fifth power. 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. So three-quarters to the fifth power is the same as three to the fifth power over four to the fifth power. And we don’t actually need to evaluate this since this is one of our options; it’s (E). 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.

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