Video: Simplifying Numerical Expressions with Negative Exponents

Which of the following is equal to (2^(−2) × 27^(−1/3) × 25^(−1/2))/4⁻³? [A] 1/240 [B] 15/16 [C] 16/15 [D] 48/5 [E] 240

03:38

Video Transcript

Which of the following is equal to two to the negative two power times 27 to the negative one-third power times 25 to the negative one-half power over four to the negative three power. A) One over 240, B) 15 over 16, C) 16 over 15, D) 48 over five, or E) 240

One thing that we notice almost immediately is that all four of these exponent values are negative. And we know that 𝑥 to the negative 𝑎 power is equal to one over 𝑥 to the 𝑎 power. That is, when you have a negative exponent in the numerator, you can move that exponent to the denominator and make it positive. And by extension, we can say one over 𝑥 to the negative 𝑎 power equals 𝑥 to the positive 𝑎 power.

In our case, it means all the values with negative exponents in the numerator should be moved to the denominator. And they will be positive. Remember, this is only possible because we are multiplying all three of these values together. We now have in the denominator two to the positive two power times 27 to the positive one-third power times 25 to the positive one-half power. And four to the negative three power can be moved to the numerator as four to the positive three power.

If we keep our numerator the same, we can calculate two squared, which is four. And then, we have to remember that 27 to the one-third power is equal to the cube root of 27. It’s saying that some value cubed equals 27. And if 𝑥 cubed equals 27, then 𝑥 equals 27 to the one-third power, or the cube root of 27. We know that three times three times three equals 27; three cubed equals 27. And that means the cube root of 27 is three. And so, in place of 27 to the one-third power, we can write three.

After that, we have the square root of 25. If 𝑥 squared equals 25, then 𝑥 equals 25 to the one-half power, which is the same thing as the square root of 25. We know the square root of 25 is five because five times itself is 25.

There is one more thing we can simplify. If we look at our four in the denominator, we could say that that is four to the first power. And since we have four cubed in the numerator and four to the first power in the denominator, we can simplify that to have four squared in the numerator. This is because when the exponents have the same base, 𝑥 to the 𝑎 power over 𝑥 to the 𝑏 power equals 𝑥 to the 𝑎 minus 𝑏 power. In our case, we had four cubed over four to the first power. And that is four to the three minus one power, four squared. Three times five is 15. Four squared is 16. So, this simplification is 16 over 15, which is option C.

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