To express [tex]\( \sqrt{32} \)[/tex] in its simplest radical form, follow these steps:
1. Factor the number inside the square root:
The number 32 can be factored into its prime factors.
[tex]\[
32 = 2 \times 16 = 2 \times 2 \times 8 = 2 \times 2 \times 2 \times 4 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5
\][/tex]
2. Simplify the square root using the properties of exponents:
Recall that the square root of a product is equal to the product of the square roots. Notably, the square root of [tex]\( a^2 \)[/tex] is [tex]\( a \)[/tex]:
[tex]\[
\sqrt{32} = \sqrt{2^5}
\][/tex]
3. Separate the even and odd powers:
We know [tex]\(2^5 = 2^4 \times 2 = (2^2)^2 \times 2 = 4^2 \times 2 \)[/tex]:
[tex]\[
\sqrt{2^5} = \sqrt{(2^2)^2 \times 2} = \sqrt{4^2 \times 2}
\][/tex]
4. Use the property [tex]\( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)[/tex]:
Now we use this property to separate out the perfect square:
[tex]\[
\sqrt{32} = \sqrt{4^2 \times 2} = \sqrt{4^2} \times \sqrt{2} = 4 \times \sqrt{2}
\][/tex]
Therefore, the simplest radical form of [tex]\( \sqrt{32} \)[/tex] is [tex]\( 4 \sqrt{2} \)[/tex]. In decimal form, this is approximately [tex]\( 5.656854249492381 \)[/tex].
