What is the square root of 32 simplified?
Understand the Problem
The question is asking for the simplified form of the square root of 32. This involves finding the prime factorization of 32 and then simplifying the square root based on that factorization.
Answer
$4\sqrt{2}$
Answer for screen readers
The simplified form of the square root of 32 is $4\sqrt{2}$.
Steps to Solve
- Find the Prime Factorization of 32
First, we need to express 32 as a product of its prime factors. The prime factorization of 32 is: $$ 32 = 2 \times 16 = 2 \times 2 \times 8 = 2 \times 2 \times 2 \times 4 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5 $$
- Rewrite the Square Root Expression
We can now rewrite the square root of 32 using its prime factorization: $$ \sqrt{32} = \sqrt{2^5} $$
- Use the Property of Square Roots
Using the property that $\sqrt{a^b} = a^{b/2}$, we can simplify the expression: $$ \sqrt{2^5} = 2^{5/2} = 2^{2} \times 2^{1/2} = 4 \sqrt{2} $$
- Final Simplified Form
Thus, the simplified form of the square root of 32 is: $$ \sqrt{32} = 4 \sqrt{2} $$
The simplified form of the square root of 32 is $4\sqrt{2}$.
More Information
The square root of a number can often be simplified by breaking it down into its prime factors. This method allows for easy simplification by applying properties of exponents and square roots.
Tips
- Ignoring Prime Factorization: A common mistake is to skip the prime factorization step. Always start by expressing the number as a product of primes.
- Not Applying Properties Correctly: Make sure to apply the property of square roots properly; $\sqrt{a^b}$ simplifies to $a^{b/2}$.
