# Question Video: Evaluating Positive Rational Expressions with Positive Integer Exponents Raised to Positive Integer Exponents Mathematics • 8th Grade

Which of the following expressions has the same value as ((1/3)³)²? [A] (1/3)⁶ [B] (1/3)⁵ [C] (1/3)⁹ [D] 2/3³ [E] (1/3)³ × (1/3)²

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### Video Transcript

Which of the following expressions has the same value as one-third cubed squared? (A) One-third to the sixth power, (B) one-third to the fifth power, (C) one-third to the ninth power, (D) two over three cubed, (E) one-third cubed multiplied by one-third squared.

We’ve been asked which of five expressions has the same value as the given expression. Now we could evaluate this expression and each of the five options by evaluating the powers and see which result in the same value. However, it will be easier to simplify the given expression using the laws of exponents.

We can recall that if we have a rational base raised to a positive integer exponent and then raise this to another positive integer exponent, then this is equivalent to raising that base to the product of those exponents. So applying this law with the base 𝑎 equal to one-third, the first exponent 𝑚 equal to three, and the second exponent 𝑛 equal to two gives one-third to the power of three times two. This of course simplifies to one-third to the power of six, which is option (A).

We should briefly consider some of the other options though as they reveal some of the commonly made mistakes when dealing with exponents.

In option (B), the answer of one-third to the fifth power is reached by adding the exponents of two and three instead of multiplying them.

In option (C), the exponent of nine comes from raising the first exponent to the power of the second; three squared is nine.

Option (D) doesn’t really reflect a common misconception surrounding exponents, but perhaps at some point the one in the numerator has been multiplied by the power of two instead of raised to it.

In option (E), the expression has been rewritten as a product of terms raised to the exponents of three and two, instead of raising the base of one-third to the power of three and then to the power of two.

Using another law of exponents — which tells us that when we multiply powers of the same base, we add the exponents — shows that this expression actually simplifies to one-third to the fifth power, which is the same as option (B).

The correct expression that has the same value as one-third cubed squared is option (A): one-third to the sixth power.