Video: AQA GCSE Mathematics Higher Tier Pack 2 • Paper 1 • Question 8

Calculate (4^(1/2))² × 9^(1/2))².

01:51

Video Transcript

Calculate four to the power of a half squared multiplied by nine to the power of a half squared.

There are two ways we can answer this question. The first is to consider what we actually mean by a power of one-half or an index of one-half. In fact, a fractional index gives a root. And the denominator of the fraction tells us which root we’re looking for. So the power of a half is a square root. And the power of a third or the index a third gives us a cube root.

In this case, both four and nine have a power of one-half. So we’re square-rooting each of these numbers inside the bracket. We could at this point work out the value of the square root since both four and nine are square numbers. However, each bracket is being squared. Squaring is the inverse or the opposite of square-rooting. So the square root of four squared is just four, and the square root of nine squared is nine. So our sum becomes four multiplied by nine, which is simply 36.

An alternative method we could’ve considered would have been to use the laws of indices. When we have a bracket with an index, we can multiply the powers. So 𝑥 to the power of 𝑎 to the power of 𝑏 is the same as 𝑥 to the power of 𝑎 multiplied by 𝑏. This means four to the power of a half all squared is four to the power of a half multiplied by two. And a half multiplied by two is simply one.

Similarly, nine to the power of a half all squared is nine to the power of a half multiplied by two, which is nine to the power of one. And in fact, anything to the power of one is just itself, so we have four and nine. Once again, we multiply these as per the original sum and we get four multiplied by nine to be 36. Either method is valid here.

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