Questions: Simplify. √32 × 4√45

Simplify.

√32 × 4√45
Transcript text: Week 8 Final Exam Question 19 of 25 (4 points) | Question Attempt: 1 of 1 Simplify. $\sqrt{32} \times 4\sqrt{45}$
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Solution

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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
  1. Simplify \(\sqrt{32}\) by expressing 32 as a product of its prime factors and taking out pairs.
  2. Simplify \(4\sqrt{45}\) by expressing 45 as a product of its prime factors and taking out pairs, then multiply by 4.
  3. 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: \[ \boxed{48\sqrt{10}} \]

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