Question Video: Evaluating an Expression with a Rational Base and a Positive Rational Exponent

Evaluate (125/343)^(2/3).

02:11

Video Transcript

Evaluate 125 over 343 to the power of two-thirds.

The first thing to note here is that our fraction, two-thirds, is an exponent or a power. And it doesn’t mean that we’re multiplying it with the fraction 125 over 343. To start, let’s take the exponent of this fraction and write it as an exponent of the numerator and an exponent of the denominator. In other words, we can use the rule that if we have a fraction 𝑥 over 𝑦 to the power of 𝑎, it’s equivalent to 𝑥 to the power of 𝑎 over 𝑦 to the power of 𝑎. So for our value, we can write our numerator as 125 to the power of two-thirds and our denominator as 343 to the power of two-thirds.

So now let’s simplify these fractional exponents of two-thirds. Recall that if we have a value 𝑥 to the power of 𝑎 over 𝑏, this is equivalent to the 𝑏th root of 𝑥 to the power of 𝑎. And therefore, on our numerator, 125 to the power of two-thirds is equivalent to the cube root of 125 squared. Our denominator is equivalent to the cube root of 343 squared.

We can notice on our numerator that this is equivalent to squaring 125 first and taking the cube root. Equally, on our denominator, we could square 343 first and then take the cube root. However, in the second form written in orange, this will have much larger numbers. Since we’re squaring 125 first and then trying to find the cube root of that. Whereas if we start by taking the cube root first and then squaring it, our values won’t get so large.

Therefore, the cube root of 125 will give us five. And since we then need to square it, we’ll have five squared on our numerator. And on our denominator, the cube root of 343 is seven, since seven times seven times seven gives us 343. And then we’ll need to square that. Evaluating our squares then will give us the final answer of 25 over 49.

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