Certainly! Let's go through the steps to find the product of [tex]\((-2 \sqrt{20k})\)[/tex] and [tex]\(5 \sqrt{8k^3}\)[/tex].
1. Express Each Term Separately:
[tex]\[
(-2 \sqrt{20k}) \quad \text{and} \quad (5 \sqrt{8k^3})
\][/tex]
2. Multiply the Coefficients and the Radicals Separately:
Coefficients:
[tex]\[
(-2) \times 5 = -10
\][/tex]
Radicals:
[tex]\[
\sqrt{20k} \times \sqrt{8k^3}
\][/tex]
3. Combine the Radicals into One Square Root:
When multiplying square roots, you can combine them under a single square root:
[tex]\[
\sqrt{20k} \times \sqrt{8k^3} = \sqrt{(20k) \times (8k^3)}
\][/tex]
4. Simplify the Expression Inside the Radical:
Multiply the terms inside the square root:
[tex]\[
20k \times 8k^3 = 160k^4
\][/tex]
5. Combine the Results:
So, we now have:
[tex]\[
-10 \sqrt{160k^4}
\][/tex]
6. Check the Matching Expressions:
From the options given:
- [tex]\(-10 \sqrt{160 k^3}\)[/tex]
- [tex]\(-10 \sqrt{160 k^4}\)[/tex]
- [tex]\(3 \sqrt{28 k^4}\)[/tex]
- [tex]\(3 \sqrt{28 k^3}\)[/tex]
The expression [tex]\(-10 \sqrt{160 k^4}\)[/tex] correctly matches our result.
Thus, the equivalent expression for the product [tex]\((-2 \sqrt{20k})(5 \sqrt{8k^3})\)[/tex] is:
[tex]\[
-10 \sqrt{160 k^4}
\][/tex]
