Question Video: Simplifying the Sum and Difference of Radicals Mathematics • 10th Grade

Video Transcript

Simplify the square root of 12 minus two times the square root of three plus four times the square root of 27.

In this question, we are asked to simplify an expression involving the sum and difference of multiple radical expressions. To do this, we should start by simplifying each of the terms. We can do this by reducing the sizes of the radicands by recalling that if 𝑎 and 𝑏 are nonnegative, then we have that the square root of 𝑎 squared times 𝑏 is equal to 𝑎 times the square root of 𝑏. This allows us to reduce the size of the radicand by finding its nontrivial square factors.

We can also note that 12 is four times three. So it is equal to two squared times three. And 27 is equal to nine times three. So it is equal to three squared times three. We cannot reduce the size of the radicand in the square root of three. Since three is prime, it only has trivial square factors. We can rewrite the expression as shown.

We can now simplify each term by using our exponent rule. First, we can set 𝑎 equal to two and 𝑏 equal to three to rewrite the square root of two squared times three as two root three. We leave the second term unchanged. In the third term, we can simplify the square root of three squared times three by using our exponent rule, with 𝑎 equal to three and 𝑏 equal to three. We see that it is equal to three root three. We need to multiply this by four and add it onto the expression.

To simplify further, we can start by recalling that 𝑎 multiplied by 𝑏 root 𝑐 is equal to 𝑎𝑏 root 𝑐. Since four times three is 12, we can rewrite the third term as 12 root three.

We can now see that all three terms share a factor of the square root of three. We can take this shared factor of root three out of the expression by using the distributive property of multiplication over addition. This allows us to just add and subtract the coefficients of root three. We can then evaluate that two minus two plus 12 is equal to 12.

Hence, we were able to show that root 12 minus two root three plus four root 27 is equal to 12 root three.