Question

Simplify: \(\sqrt{32}=\)_____ ;\(-\sqrt{45}=\)_____

Short answer

\(\sqrt{32} = 4\sqrt{2}\); \(-\sqrt{45} = -3\sqrt{5}\)

Step-by-step solution

Simplify \( \sqrt{32} \)

Find the prime factorization of 32. The factors are 32 = 2 * 2 * 2 * 2 * 2, or written as powers of 2, 32 = 2^5. To simplify a square root, factor out pairs of prime factors: \(\sqrt{32} = \sqrt{2^5} = \sqrt{(2^2) \times (2^2) \times 2} = 2 \times 2 \times \sqrt{2} = 4\sqrt{2}.\)

Simplify \( -\sqrt{45} \)

Find the prime factorization of 45. The factors are 45 = 3 * 3 * 5. To simplify a square root, factor out pairs of prime factors: \(-\sqrt{45} = -\sqrt{\(3^2 * 5\)} = -3 \sqrt{5}.\)

Key concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself.
To prime factorize a number, keep dividing it by the smallest prime numbers (2, 3, 5, 7, 11, etc.) until only 1 is left.
For example, in the original exercise, to find the prime factorization of 32, you divide by 2 (the smallest prime number) since 32 is even. You continue:
32 ÷ 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 = 2
2 ÷ 2 = 1
Thus, 32 can be written as 2 * 2 * 2 * 2 * 2 or 2^5.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. The symbol for the square root is \( \sqrt{} \).
For example, the square root of 9 is 3 because 3 * 3 = 9.
Not all numbers have simple whole number square roots. When dealing with such numbers, we need to simplify the square root using prime factorization.
In the exercise, for \( \sqrt{32} \), we have to use its prime factors. The square root of a number multiplied by itself will result in the original number.
So \( \sqrt{32} = \sqrt{(2^2 \times 2^2 \times 2)} \), where 2 appears in pairs. We take one out from each pair:
\( \sqrt{32} = 2 \times 2 \times \sqrt{2} \), which simplifies to 4\( \sqrt{2} \).
Understanding square roots and their properties helps simplify them for easier handling in mathematical problems.

Simplification Steps

Simplifying square roots involves a few clear steps. Let's walk through them with an example from the exercise:
1. Find the prime factorization of the number under the square root. For 32, it is 2^5.
2. Group the prime factors into pairs. From the example, we have (2^2) \times (2^2) \times 2.
3. For each pair, take one factor out of the square root. This gives us 2 \times 2 = 4 on the outside.
4. Multiply the factors outside the square root together. For \( \sqrt{32} \), this resulted in 4. Keep any remaining factors inside the square root.
5. Combine the simplified factors. The 2 left inside the root stays inside, so the final simplified form is 4\( \sqrt{2} \).
Similarly, for negative square roots like \( -\sqrt{45} \), follow the same steps but keep the negative sign:
Prime factorize 45: 3^2 \times 5.
Take one factor out for each pair: 3 from 3^2.
The simplified form: -3\( \sqrt{5} \).
Understanding these simplification steps ensures you handle square roots efficiently.