What is the square root of 72 simplified?

Understand the Problem

The question is asking for the simplified form of the square root of 72. To solve it, we'll factor 72 into its prime factors and then simplify the square root based on those factors.

Answer

$6\sqrt{2}$

Steps to Solve

Factor the number
We start by finding the prime factors of 72. We can do this by dividing 72 by the smallest prime numbers.
- $72 \div 2 = 36$
- $36 \div 2 = 18$
- $18 \div 2 = 9$
- $9 \div 3 = 3$
- $3 \div 3 = 1$
Thus, the prime factorization of 72 is:

$$ 72 = 2^3 \times 3^2 $$
Identify perfect squares
In the prime factorization, we need to identify the perfect squares. In this case:
- $2^3$ can be split into $2^2 \times 2 = 4 \times 2$
- $3^2$ is already a perfect square.
Rewrite the square root
Using the identified perfect squares, we can rewrite the square root of 72:

$$ \sqrt{72} = \sqrt{2^2 \times 2 \times 3^2} $$
Simplify the square root
Now we can separate the perfect squares from the non-perfect squares:

$$ \sqrt{72} = \sqrt{(2^2)(3^2)(2)} = \sqrt{4} \times \sqrt{9} \times \sqrt{2} $$

Simplifying further gives us:

$$ \sqrt{4} = 2 \quad \text{and} \quad \sqrt{9} = 3 $$

Therefore:

$$ \sqrt{72} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2} $$

The simplified form of the square root of 72 is $6\sqrt{2}$.

More Information

The process of simplifying square roots involves breaking down numbers into their prime factors. This helps us to identify perfect squares, which can be evaluated easily, leading to a more understandable answer. Additionally, recognizing that $6\sqrt{2}$ demonstrates the relationship between whole numbers and their radical components is an essential math skill.

Tips

Forgetting to include all prime factors when factorizing.
Not recognizing perfect squares can lead to failure in simplification.
Confusing the order of operations, particularly when separating the square root.

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