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.
