Questions: Simplify the following: √567

Transcript text: Simplify the following: $\sqrt{567}$

Solution

Solution Steps

To simplify the square root of 567, we need to find the prime factorization of 567 and identify any perfect squares. We can then simplify the square root by taking the square root of the perfect square factors.

Step 1: Prime Factorization of 567

To simplify $\sqrt{567}$, we first find the prime factorization of 567. The prime factors of 567 are $3^4 \times 7$.

Step 2: Identify Perfect Squares

From the prime factorization, we can identify the perfect square. The factor $3^4$ is a perfect square because it can be expressed as $(3^2)^2$.

Step 3: Simplify the Square Root

We can simplify $\sqrt{567}$ by taking the square root of the perfect square factor: \[ \sqrt{567} = \sqrt{3^4 \times 7} = \sqrt{(3^2)^2 \times 7} = 3^2 \times \sqrt{7} = 9\sqrt{7} \]