Video: AQA GCSE Mathematics Higher Tier Pack 5 β€’ Paper 2 β€’ Question 5

Fully simplify (5π‘₯³𝑦⁴)Β³.

04:26

Video Transcript

Fully simplify five π‘₯ cubed 𝑦 to the power of four all cubed.

There are a couple of different ways that we could approach this question. The first way is to write this expression out in a slightly more longhand way. Five π‘₯ cubed 𝑦 to the four all cubed means we’re multiplying five π‘₯ cubed 𝑦 to the four by itself three times. We have five π‘₯ cubed 𝑦 to the four multiplied by five π‘₯ cubed 𝑦 to the four multiplied by five π‘₯ cubed 𝑦 to the four.

It doesn’t matter what order we multiply in. If you think about numbers, for example, two multiplied by six is 12, and so is six multiplied by two. So we can reorder the parts of this product to bring the numbers together, the π‘₯ terms together, and the 𝑦 terms together. To give five multiplied by five multiplied by five multiplied by π‘₯ cubed multiplied by π‘₯ cubed multiplied by π‘₯ cubed, multiplied by 𝑦 to the four multiplied by 𝑦 to the four multiplied by 𝑦 to the four. Five multiplied by five is 25, and multiplying by five again gives 125. So we can simplify the number part of this expression in this way.

To simplify the letter parts, we need to recall one of our rules of powers. If we’re multiplying together two terms with the same base β€” in this case, that’s π‘₯ β€” then we add the powers together. π‘₯ to the power of π‘Ž multiplied by π‘₯ to the power of 𝑏 is equal to π‘₯ to the power of π‘Ž plus 𝑏.

To see why this is, we can again think of a longhand version of writing this out. π‘₯ to the power of π‘Ž would be π‘₯ times π‘₯ times π‘₯ π‘Ž times. And π‘₯ to the power of 𝑏 is π‘₯ times π‘₯ times π‘₯ 𝑏 times. So in total, we multiply π‘₯ together π‘Ž plus 𝑏 times. So π‘₯ cubed multiplied by π‘₯ cubed multiplied by π‘₯ cubed is equal to π‘₯ to the power of three plus three plus three. We add all three powers together. Three plus three plus three is nine. So this part of the expression simplifies to π‘₯ to the power of nine.

We can simplify the 𝑦 part of this expression in the same way. It becomes 𝑦 to the power of four plus four plus four, which is equal to 𝑦 to the power of 12. Remember, we’re multiplying these three parts together. So we can remove the multiplication signs. And it gives 125π‘₯ to the power of nine 𝑦 to the power of 12.

That’s our first method. But there is actually another way which is a little bit easier. And to see this, we need to recall one of our other index laws. It’s called the power law. And it tells us that if we’re raising some number or letter, in this case π‘₯, to a power and then to another power, we can multiply the powers together. So π‘₯ to the power of π‘Ž and then to the power of 𝑏 is equal to π‘₯ to the power of π‘Žπ‘.

We’re also going to apply the rule that if we have a product raised to a power β€” so that’s π‘₯𝑦 to the power of π‘Ž β€” then this is the same as the product of each of those letters individually raised to the power. π‘₯𝑦 all to the power of π‘Ž is equal to π‘₯ to the power of π‘Ž multiplied by 𝑦 to the power of π‘Ž. What this means then is we can take each part of this expression, individually raise it to the power on the outside of the bracket β€” that’s three β€” and find the product of these values.

So we have five cubed multiplied by π‘₯ cubed, cubed, multiplied by 𝑦 to the power of four, cubed. Five cubed is 125. And then we’ll use the first rule that we wrote down in this method to simplify the powers of π‘₯ and 𝑦. Remember, if we’re raising a letter to a power and then another power, we multiply the powers together.

So π‘₯ cubed to the power of three is π‘₯ to the power of three times three. And 𝑦 to the power of four to the power of three is 𝑦 to the power of four times three. Three threes are nine, and four threes are 12. So we have 125 multiplied by π‘₯ to the power of nine multiplied by 𝑦 to the power of 12. We can remove the multiplication signs. And we see that it gives the same answer as we found using our first method. The fully simplified form of five π‘₯ cubed 𝑦 to the four all cubed is 125π‘₯ to the power of nine 𝑦 to the power of 12.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.