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Chapter 7: Problem 37

Simplify the products. Give exact answers. $$2 \sqrt{7 a} \cdot 3 \sqrt{2 a}$$

Short Answer

Expert verified
The simplified product is \(6a \sqrt{14}\).

Step by step solution

01

- Multiply the coefficients

Multiply the numerical coefficients outside the square roots. Here the coefficients are 2 and 3.So, calculate: \[2 \times 3 = 6\]
02

- Multiply the radicals under the square roots

Next, multiply the terms inside the square roots. Here, the terms are \(\sqrt{7a}\) and \(\sqrt{2a}\).So, calculate: \[\sqrt{7a} \times \sqrt{2a} = \sqrt{(7a) \times (2a)}\]
03

- Simplify under the square root

Multiply inside the square root to simplify further:\[\sqrt{(7a) \times (2a)} = \sqrt{14a^2}\]
04

- Simplify the square root

The expression under the square root can be simplified since \(a^2\) is a perfect square. So take the square root of \(a^2\):\[\sqrt{14a^2} = \sqrt{14} \times \sqrt{a^2} = \sqrt{14} \times a = a \sqrt{14}\]
05

- Combine the simplified parts

Now combine the simplified results from Step 1 and Step 4:\[6 \times a \sqrt{14} = 6a \sqrt{14}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplying Coefficients
When simplifying radical expressions that involve multiplication, the first step is to focus on the coefficients, which are the numbers outside the square roots. In our exercise, the coefficients are 2 and 3. Multiplying these two numbers is straightforward:
  • 2 times 3 equals 6.
    So, we have:

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