Question
What is the product of the following in simplest radical form?
8sqrt(5) * 3sqrt(32)
8sqrt(5) * 3sqrt(32)
Answers
Answered by
GPT-4o mini
To find the product of \( 8\sqrt{5} \) and \( 3\sqrt{32} \), we start by multiplying the coefficients and the radicals separately.
1. **Multiply the coefficients**:
\[
8 \times 3 = 24
\]
2. **Multiply the radicals**:
\[
\sqrt{5} \times \sqrt{32} = \sqrt{5 \times 32}
\]
3. **Calculate \(5 \times 32\)**:
\[
5 \times 32 = 160
\]
So, we have:
\[
\sqrt{5} \times \sqrt{32} = \sqrt{160}
\]
4. **Now combine the results**:
\[
8\sqrt{5} \times 3\sqrt{32} = 24\sqrt{160}
\]
5. **Simplify \(\sqrt{160}\)**:
We first factor \(160\):
\[
160 = 16 \times 10 = 16 \times (2 \times 5)
\]
Since \(16\) is a perfect square, we can simplify:
\[
\sqrt{160} = \sqrt{16 \times 10} = \sqrt{16} \times \sqrt{10} = 4\sqrt{10}
\]
6. **Substituting back**:
We now substitute \(\sqrt{160}\) back into our expression:
\[
24\sqrt{160} = 24 \times 4\sqrt{10} = 96\sqrt{10}
\]
Thus, the product \(8\sqrt{5} \times 3\sqrt{32}\) in simplest radical form is:
\[
\boxed{96\sqrt{10}}
\]
1. **Multiply the coefficients**:
\[
8 \times 3 = 24
\]
2. **Multiply the radicals**:
\[
\sqrt{5} \times \sqrt{32} = \sqrt{5 \times 32}
\]
3. **Calculate \(5 \times 32\)**:
\[
5 \times 32 = 160
\]
So, we have:
\[
\sqrt{5} \times \sqrt{32} = \sqrt{160}
\]
4. **Now combine the results**:
\[
8\sqrt{5} \times 3\sqrt{32} = 24\sqrt{160}
\]
5. **Simplify \(\sqrt{160}\)**:
We first factor \(160\):
\[
160 = 16 \times 10 = 16 \times (2 \times 5)
\]
Since \(16\) is a perfect square, we can simplify:
\[
\sqrt{160} = \sqrt{16 \times 10} = \sqrt{16} \times \sqrt{10} = 4\sqrt{10}
\]
6. **Substituting back**:
We now substitute \(\sqrt{160}\) back into our expression:
\[
24\sqrt{160} = 24 \times 4\sqrt{10} = 96\sqrt{10}
\]
Thus, the product \(8\sqrt{5} \times 3\sqrt{32}\) in simplest radical form is:
\[
\boxed{96\sqrt{10}}
\]
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