Video Transcript
Write the square root of eight plus the square root of two in the form π times the square root of two, where π is an integer.
In this question, we are asked to rewrite the sum of two radicals into a single radical. We can see that we want the radicand in the rewritten expression to be equal to two. To do this, we can recall that we can use the fact that for any nonnegative real numbers π and π, we have that the square root of π squared times π is equal to π times the square root of π. This result allows us to reduce the size of the radicand in the first term by noting that eight is equal to two squared times two. Therefore, if we set π equal to two and π equal to two, we have that the square root of eight is equal to two times the square root of two, giving us the following expression.
We can now see that we are adding two terms together that share a factor of the square root of two. We can combine these two terms by recalling the distributive property of multiplication over addition, which tells us that for any real numbers π, π, and π, we have that ππ plus ππ is equal to π plus π multiplied by π. Noting that the square root of two is equal to one times the square root of two, we can apply this result with π equal to two, π equal to one, and π equal to the square root of two to obtain two plus one multiplied by the square root of two. Finally, we can add the coefficients to get that the square root of eight plus the square root of two is equal to three times the square root of two.
