square root of 45 simplfy by factoring

Factor 45 into 9 times 5

Since 9 is a perfect square, you can take its square root and you will have the 5 left under the square root sign.

To simplify the square root of 45 by factoring, we need to break down 45 into its prime factors and then simplify the square root expression.

Step 1: Factorize 45:
The prime factorization of 45 is 3 * 3 * 5.

Step 2: Simplify the square root expression:
√45 = √(3 * 3 * 5)

Using the property of square roots (√a * √b = √(a * b)), we can rewrite the expression as:
√45 = √(3 * 3 * 5) = √(3 * 3) * √5

Simplifying further:
√45 = (3 * 3) * √5 = 9√5

Therefore, the simplified form of the square root of 45 by factoring is 9√5.

To find the square root of 45 and simplify it by factoring, let's break down the process step by step:

Step 1: Prime Factorization
First, we'll find the prime factors of 45. To do this, we divide 45 by the smallest prime number, which is 2. If 45 is divisible evenly by 2, we divide it by 2 and continue this process until we can no longer divide it by 2.

45 ÷ 2 = 22 remainder 1 (not divisible by 2)
Next, we try dividing 45 by the next prime number, which is 3.

45 ÷ 3 = 15 (no remainder)
Since 15 is still divisible by 3, we continue this process.

15 ÷ 3 = 5 (no remainder)

Now, we cannot divide 5 by any prime number larger than 1, so the prime factorization of 45 is 3 * 3 * 5, or simply 3^2 * 5.

Step 2: Simplify the Square Root
Since we are looking for the square root (√) of 45, we can simplify it using the prime factorization we found in step 1.

√(45) = √(3^2 * 5)

According to the properties of square roots, we can split a product inside a square root into separate square roots.

√(3^2 * 5) = √(3^2) * √(5)

Simplifying further, we get:

= 3 * √(5)

So, the simplified form of the square root of 45, factored as 3^2 * 5, is 3√(5).