Transcript text: Week 8 Final Exam Question 19 of 25 (4 points) | Question Attempt: 1 of 1
Simplify.
$\sqrt{32} \times 4\sqrt{45}$
Solution
Solution Steps
To simplify the expression \(\sqrt{32} \times 4\sqrt{45}\), we can first simplify each square root separately. The square root of a number can be simplified by factoring the number into its prime factors and then taking out pairs of prime factors as a single factor outside the square root. After simplifying each square root, we multiply the results together.
Solution Approach
Simplify \(\sqrt{32}\) by expressing 32 as a product of its prime factors and taking out pairs.
Simplify \(4\sqrt{45}\) by expressing 45 as a product of its prime factors and taking out pairs, then multiply by 4.
Multiply the simplified results from steps 1 and 2.
Step 1: Simplifying \(\sqrt{32}\)
To simplify \(\sqrt{32}\), we can factor 32 as follows:
\[
32 = 16 \times 2 = 4^2 \times 2
\]
Thus, we have:
\[
\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}
\]
Step 2: Simplifying \(4\sqrt{45}\)
Next, we simplify \(4\sqrt{45}\). We can factor 45 as:
\[
45 = 9 \times 5 = 3^2 \times 5
\]
Therefore, we have:
\[
4\sqrt{45} = 4\sqrt{9 \times 5} = 4\sqrt{9} \times \sqrt{5} = 4 \times 3\sqrt{5} = 12\sqrt{5}
\]
Step 3: Multiplying the Results
Now, we multiply the simplified results from Steps 1 and 2:
\[
\sqrt{32} \times 4\sqrt{45} = (4\sqrt{2}) \times (12\sqrt{5}) = 48\sqrt{10}
\]
Final Answer
The final simplified result is:
\[
\boxed{48\sqrt{10}}
\]