Video: AQA GCSE Mathematics Higher Tier Pack 5 • Paper 1 • Question 27

Work out ∛27 × 2⁻². Give your answer as a decimal.

03:58

Video Transcript

Work out the cube root of 27 multiplied by two to the power of negative two. Give your answer as a decimal.

Let’s consider the two parts of this product separately to begin with. We need to work out the cube root of 27 and also two to the power of negative two. The cube root of 27 just means the number that we multiply together — that’s by itself three times — in order to give 27.

It’s a good idea to learn the first few cube numbers off by heart. For example, one multiplied by one multiplied by one is one. So one is a cube number. Two multiplied by two is four and multiplying by two again gives eight. So eight is also a cube number. Three multiplied by three is nine and multiplying by three again gives 27. so 27 is a cube number. 27 can, therefore, be written as three cubed, which in turn means that the cube root of 27 is equal to three. It will be sensible to learn the next couple of cube numbers as well. Four cubed is 64, five cubed is 125, and six cubed is 216.

Now, let’s consider the second part of this product: two to the power of negative two. And in order to work this out, we need to recall what is meant by a negative power. A negative power defines a reciprocal. The general rule is that if we have 𝑥 to the power of negative 𝑎, then this is equal to one over 𝑥 to the power of 𝑎.

So two to the power of negative two is equal to one over two to the power of two or one over two squared. Two squared is equal to four. So we have that two to the power of negative two is equal to a quarter. You can see why this definition of negative powers works if you consider the powers of two. Two cubed is eight, two squared is four, and two to the power of one is two.

Each time, when we decrease the power by one, we divide by two. So eight divided by two gives four, four divided by two gives two. Two to the power of zero then is equal to one. To get to two to the power of negative one, we have to divide one by two which gives a half. And then to find two to the power of negative two, we have to divide by two again. A half divided by two is one-quarter, which we can also write as one over two squared.

So now we’ve worked each of these powers out individually, we can return to working out their product. We have that the cube root of 27 multiplied by two to the power of negative two is equal to three multiplied by one-quarter. This is equal to three-quarters, which you may be able to see straightaway. But if not, you can write the integer three as the fraction of three over one and then multiply the two fractions together by multiplying the numerators to give three and multiplying the denominators to give four.

However, the question asked us to give our answer as a decimal. So finally, we need to convert three-quarters into a decimal. This is equal to 0.75. Now, this is one of the common decimal fraction equivalents that you should know off by heart. But if you didn’t know that three-quarters was equal to 0.75, you could work this out using a short division.

We set up our short division with three underneath the bus stop division sign and four on the outside. We’ll include the decimal point and add some extra zeros after it. There are no fours in three. So we put a zero and carry the three into the next column. Seven fours are 28. So there are seven fours in 30 with a remainder of two. Five fours are 20 with no remainder. So we found by division that three divided by four is equal to 0.75.

We’ve worked out then that the cube root of 27 multiplied by two to the power of negative two as a decimal is equal to 0.75.

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