Answer :
To solve the problem and find which choice is equivalent to the expression [tex]\(\sqrt{32} - \sqrt{2}\)[/tex], let's simplify the expression step by step.
1. Simplify [tex]\(\sqrt{32}\)[/tex]:
- First, find the prime factorization of 32:
[tex]\[
32 = 16 \times 2 = 4^2 \times 2
\][/tex]
- So, [tex]\(\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \cdot \sqrt{2}\)[/tex].
- Since [tex]\(\sqrt{16} = 4\)[/tex], we have:
[tex]\[
\sqrt{32} = 4\sqrt{2}
\][/tex]
2. Subtraction:
- Now, subtract [tex]\(\sqrt{2}\)[/tex] from [tex]\(4\sqrt{2}\)[/tex]:
[tex]\[
4\sqrt{2} - \sqrt{2} = (4 - 1)\sqrt{2} = 3\sqrt{2}
\][/tex]
With this simplification, the expression [tex]\(\sqrt{32} - \sqrt{2}\)[/tex] simplifies to [tex]\(3\sqrt{2}\)[/tex].
3. Compare with the options:
- Option A: [tex]\(16\sqrt{2}\)[/tex]
- Option B: [tex]\(3\sqrt{2}\)[/tex]
- Option C: 4
- Option D: [tex]\(\sqrt{30}\)[/tex]
The correct choice that matches our simplification is Option B: [tex]\(3\sqrt{2}\)[/tex].
1. Simplify [tex]\(\sqrt{32}\)[/tex]:
- First, find the prime factorization of 32:
[tex]\[
32 = 16 \times 2 = 4^2 \times 2
\][/tex]
- So, [tex]\(\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \cdot \sqrt{2}\)[/tex].
- Since [tex]\(\sqrt{16} = 4\)[/tex], we have:
[tex]\[
\sqrt{32} = 4\sqrt{2}
\][/tex]
2. Subtraction:
- Now, subtract [tex]\(\sqrt{2}\)[/tex] from [tex]\(4\sqrt{2}\)[/tex]:
[tex]\[
4\sqrt{2} - \sqrt{2} = (4 - 1)\sqrt{2} = 3\sqrt{2}
\][/tex]
With this simplification, the expression [tex]\(\sqrt{32} - \sqrt{2}\)[/tex] simplifies to [tex]\(3\sqrt{2}\)[/tex].
3. Compare with the options:
- Option A: [tex]\(16\sqrt{2}\)[/tex]
- Option B: [tex]\(3\sqrt{2}\)[/tex]
- Option C: 4
- Option D: [tex]\(\sqrt{30}\)[/tex]
The correct choice that matches our simplification is Option B: [tex]\(3\sqrt{2}\)[/tex].