Question Video: Simplifying the Quotient of Two Square Roots Mathematics • 10th Grade

Express (√10)/(√2) in its simplest form.

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Video Transcript

Express the square root of 10 over the square root of two in its simplest form.

In this question, we are asked to rewrite an expression involving the quotient of two square roots in its simplest form.

There are two ways that we can do this.

The first way of answering this question is to recall that we can simplify the quotient of roots by using the quotient rule for square roots. This tells us that if 𝑎 is a nonnegative real number and 𝑏 is a positive real number, then the square root of 𝑎 over the square root of 𝑏 is equal to the square root of 𝑎 over 𝑏. We can apply this result with 𝑎 equal to 10 and 𝑏 equal to two to see that the square root of 10 over the square root of two is equal to the square root of 10 over two. We can then calculate that 10 over two is equal to five, giving us the square root of five. Since five is a prime number, we cannot simplify this expression any further.

Another way of answering this question is to try to rewrite the numerator of the expression. We can do this by recalling that for any nonnegative real numbers 𝑎 and 𝑏, we have that the square root of 𝑎 times 𝑏 is equal to the square root of 𝑎 multiplied by the square root of 𝑏. We can apply this to the numerator of the expression by noting that 10 is equal to five times two. Therefore, we can set 𝑎 equal to five and 𝑏 equal to two to get the square root of five multiplied by the square root of two over the square root of two. We can then cancel the shared factor of the square root of two in the numerator and denominator to once again show that the square root of 10 over the square root of two is equal to the square root of five.