How do you simplify the square root of 72?

To simplify the square root of 72, we can break it down into its prime factors and look for perfect squares. This process will help us express the square root in a simpler form.

Breaking Down the Number

First, let's find the prime factorization of 72. We can do this by dividing 72 by the smallest prime numbers until we reach 1:

• 72 ÷ 2 = 36
• 36 ÷ 2 = 18
• 18 ÷ 2 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

So, the prime factorization of 72 is:

72 = 2 × 2 × 2 × 3 × 3 or 72 = 2³ × 3²

Identifying Perfect Squares

Next, we look for perfect squares among the factors. A perfect square is a number that can be expressed as the square of an integer. In our case, we have:

• 2² = 4 (which is a perfect square)
• 3² = 9 (which is also a perfect square)

From the factorization, we can see that we have one pair of 2s (2²) and one pair of 3s (3²). We can take the square root of these pairs out of the radical.

Taking the Square Root

Now, we can simplify the square root of 72:

√72 = √(2² × 3² × 2) = √(4 × 9 × 2)

Taking the square roots of the perfect squares gives us:

√72 = 2 × 3 × √2 = 6√2

Final Result

Thus, the simplified form of the square root of 72 is:

√72 = 6√2

This means that while the square root of 72 is approximately 8.49, in its simplest radical form, it is expressed as 6 times the square root of 2. This method of simplifying square roots can be applied to other numbers as well, making it a useful skill in mathematics.