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Math  /  Numbers & Operations

QuestionSimplify. 45\sqrt{45} □\square submit

Studdy Solution

STEP 1

1. We are simplifying the square root of 45 45 .
2. The goal is to express 45 \sqrt{45} in its simplest radical form.

STEP 2

1. Identify the prime factors of the number under the square root.
2. Simplify the square root by grouping the factors.

STEP 3

Find the prime factorization of 45 45 .
45=3×15 45 = 3 \times 15 15=3×5 15 = 3 \times 5
Thus, the prime factorization of 45 45 is:
45=3×3×5 45 = 3 \times 3 \times 5

STEP 4

Simplify the square root by grouping the factors.
Since 45=32×5 45 = 3^2 \times 5 , we can write:
45=32×5 \sqrt{45} = \sqrt{3^2 \times 5}
Apply the property of square roots that allows us to separate the factors:
45=32×5 \sqrt{45} = \sqrt{3^2} \times \sqrt{5}

STEP 5

Simplify further by evaluating the square root of the perfect square.
32=3 \sqrt{3^2} = 3
Therefore:
45=3×5 \sqrt{45} = 3 \times \sqrt{5}
The simplified form of 45 \sqrt{45} is:
35\boxed{3\sqrt{5}}

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