Video Transcript
Given that 𝑥 equals the fifth root of 32 subtract the fourth root of 625 plus the square root of 64, find 𝑥.
In this question, in order to find the value of 𝑥, we’ll need to work out the value of all of these different component parts on the right-hand side of the equation. Let’s take each of these terms in turn and see if we can calculate their value.
In order to find the fifth root of 32, what we’re really trying to find is what value of 𝑎 written five times and multiplied would give us a value of 32. We can say that this value of 𝑎 is not going to be one because one multiplied by one however many times would still give us a value of one. If we try a value of two, well, two multiplied by two gives us four, and another two multiplied by two gives us four, and four times four is 16, and 16 multiplied by two does indeed give us 32. This means that two to the power of five is 32. And therefore, the fifth root of 32 is two.
Next, let’s have a look at solving the fourth root of 625. This time, we’re looking for a value, let’s call it 𝑏, written four times and multiplied that would give us 625. We know that 𝑏 isn’t going to be one. And we might even realize that 𝑏 can’t be two either because 625 is an odd number. Two multiplied by two as many times would always give us an even number. So, what about a value of three? Well, three times three is nine. And we have another three times three, which is nine. And nine times nine gives us a value of 81. So, we know that our value is not three.
We could choose the value of four next, but we know that once again four is an even number and the value of 625 that we’re looking for is an odd number. In order to try a value of five, we can see that five times five gives us 25. Multiplying by another five would give us 125. And multiplying by the remaining five would give a value of 625. And that’s the value that we’re looking for. So now, we know that the fourth root of 625 is five.
Finally, let’s look at the square root of 64. We should remember that if we’re trying to find the square root of 64, we’re asking, what value multiplied by itself would give us 64? And that would be eight since eight multiplied by eight is 64. Now that we’ve found each term on the right-hand side in a simplified form, we can find the value of 𝑥.
So, we calculate the fifth root of 32, which is two, subtract the fourth root of 625, which is five, plus the square root of 64, which is eight. We’ll need to apply the order of operations here, which tells us that when we have addition and subtraction, then we perform those in order from left to right. Two subtract five gives us three. And then, adding eight will give us the value of five. Therefore, the answer to the question is that 𝑥 equals five.
