Evaluate the square root of the square root of 625.
To start solving this double square root, we’ve got the square root of the square root of 625, what we’ll need to do is to solve the first inner square root first and then square root our answer again. It can be helpful to think of this in reverse if we say that the answer to this would be 𝑥. And what we’re really thinking here is what value 𝑥 so that when we square it and then Square it again will give us 625.
If you have a calculator, it’s very easy to solve. You just simply plug in the square root of 625 and then find the square root of your answer. But let’s imagine we don’t have a calculator here. One way to solve this without a calculator is to use a trial and improvement method, where we try different values of 𝑥 and square and square again to see if we can get to 625. If we take a first trial of 𝑥 equals one, then one squared and squared again will give one. The next trial of two. Two squared and squared again would give 16. And we could continue this until we get to 625.
So another method we can use is to think of factors of 625. Since we know that 625 is a square number, we’re looking for a pair of factors where the factors are the same number. Since 25 times 25 will give us 625. We can make this as a factor pair. It’s 25 squared. Then we need to take another square root. So we’re looking for a factor pair for 25. And that will be five times five or five squared.
And now we know that if we take the number five and square it, that gives us 25, and squaring that will give us 625. So that means that the square root of the square root of 625 is five.