Video: Pack 5 • Paper 1 • Question 7

Pack 5 • Paper 1 • Question 7

03:47

Video Transcript

Evaluate four-ninths to the power of three over two.

First, we will need to recall the exponent power rule. This tells us that 𝑎 to the power of 𝑚 times 𝑛 is equal to 𝑎 to the power of 𝑚 to the power of 𝑛.

And now since we can write three over two as one-half times three, we can use the exponent power rule in order to write four-ninths to the power of three over two is equal to four-ninths to the power of a half all to the power of three.

Next, we will be using the exponent product rule. And this tells us that 𝑎 timesed by 𝑏 all to the power of 𝑚 is equal to 𝑎 to the 𝑚 timesed by 𝑏 to the 𝑚. And now we know that four-ninths can be written as four timesed by one over nine. And so we can write four-ninths to the power of a half as four to the power of a half timesed by one over nine to the power of a half. And this is also equal to four to the power of a half over nine to the power of a half. And we get that four-ninths to the power of three over two is also equal to four to the power of a half over nine to the power of a half all cubed.

Now we can use the fact that 𝑥 to the power of a half is simply the square root of 𝑥. And so four to the power of a half is going to be equal to the square root of four. And nine to the power of a half is gonna be equal to the square root of nine. And we get that this is equal to the square root of four over the square root of nine all cubed.

And now we are able to take the square root of four and nine, leaving us with two over three all cubed. Now it would not be wrong here to con- also consider the negative square root. However, we will only consider the positive one.

Next, we can again use the exponent power rule to get that this is equal to two cubed over three cubed. And then since two cubed is eight and three cubed is 27, this gives us a final answer of eight over 27.

Evaluate three over two to the power of minus three. First, we will use the exponent power rule, which, if we remember, tells us that 𝑎 to the power of 𝑚𝑛 is equal to 𝑎 to the power of 𝑚 times 𝑎 to the power of 𝑛. And since minus three is equal to minus one times three, we can write this as three over two to the power of minus one all cubed.

Next, we will use the fact that 𝑥 to the power of minus one is equal to one over 𝑥. And so, therefore, three over two to the power of minus one is equal to two over three. And therefore, three over two to the power of minus three is equal to two over three cubed.

Next, we will use the exponent product rule, which, if we remember, tells us that 𝑎 times 𝑏 to the power of 𝑚 is equal to 𝑎 to the power of 𝑚 timesed by 𝑏 to the power of 𝑚. Therefore, we get that two over three cubed is equal to two cubed over three cubed. Then since two cubed is equal to eight and three cubed is equal to 27, we get a final answer here of eight over 27.

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