QuestionSimplify the square root: $\sqrt{120}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to simplify the square root of120. . To simplify a square root, we need to find the prime factors of the number under the root.

STEP 2

First, we need to find the prime factors of120. We can do this by dividing120 by the smallest prime number (2) and continue this process until we have only prime numbers.

STEP 3

Start the prime factorization of120.
$120 =2 \times60$

STEP 4

Continue the prime factorization.
$60 =2 \times30$

STEP 5

Continue the prime factorization.
$30 =2 \times15$

STEP 6

Continue the prime factorization.
$15 =3 \times5$

STEP 7

Now we have all the prime factors of120. They are2,2,2,3, and5.

STEP 8

To simplify the square root, we group the prime factors in pairs. Each pair of the same number can come out of the square root as that number.

STEP 9

Group the prime factors in pairs.
$\sqrt{120} = \sqrt{2 \times2 \times2 \times2 \times3 \times5} = \sqrt{(2 \times2) \times (2 \times2) \times3 \times5}$

STEP 10

implify the square root by bringing out the pairs of numbers.
$\sqrt{(2 \times2) \times (2 \times2) \times3 \times5} =2 \times2 \times \sqrt{3 \times5}$

STEP 11

Calculate the multiplication outside of the square root and simplify the numbers inside the square root.
$\times \times \sqrt{3 \times5} =4 \times \sqrt{15}$ So, the simplified form of $\sqrt{120}$ is $4\sqrt{15}$ .

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