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Chapter 7: Problem 35

Simplify the products. Give exact answers. $$2 \sqrt{5} \cdot 3 \sqrt{10}$$

Short Answer

Expert verified
\(30 \sqrt{2} \)

Step by step solution

01

Multiply the coefficients

First, multiply the numerical coefficients outside the square roots. So, multiply 2 by 3. \(2 \times 3 = 6\)
02

Multiply the radicands

Next, multiply the numbers inside the square roots (the radicands). Multiply \( \sqrt{5} \times \sqrt{10} \).This results in\( \sqrt{5 \times 10} = \sqrt{50} \)
03

Combine the results

Now, combine the result from Step 1 with the result from Step 2 to get\(6 \sqrt{50} \).
04

Simplify the square root

Simplify \( \sqrt{50} \) by breaking it into prime factors:\(50 = 25 \times 2 = 5^2 \times 2 \). Therefore,\( \sqrt{50}= \sqrt{5^2 \times 2}=5 \sqrt{2} \).So the expression becomes: \(6 \times 5 \sqrt{2} \).
05

Final multiplication

Finally, multiply the coefficients obtained in Step 4: \(6 \times 5 \).The final simplified product is: \(30 \sqrt{2} \).

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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, one of the first steps is to multiply the coefficients. Coefficients are the numbers located outside the square root symbol. For the example given, we have the coefficients 2 and 3 in the expression \(2 \sqrt{5} \cdot 3 \sqrt{10}\).

To find the product of the coefficients, simply multiply these numbers together:
  • Multiply 2 by 3: \(2 \times 3 = 6\)
  • The resulting coefficient for the next step is 6.
Multiplying Radicands
After multiplying the coefficients, the next step is to multiply the radicands. Radicands are the numbers inside the square root symbol. In our example, the radicands are 5 and 10 in \( \sqrt{5} \cdot \sqrt{10}\).

To multiply radicands, use the multiplication property of square roots: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). For our operation:
  • Multiply 5 by 10: \(\sqrt{5 \cdot 10} = \sqrt{50}\)
  • The resulting radicand is \( \sqrt{50}\).

You combine this with the coefficient from the previous step: 6 \( \sqrt{50}\).
Simplifying Square Roots
Prime Factorization
Prime factorization is an essential tool in simplifying square roots. It involves breaking down a number into the product of its prime numbers. For \( \sqrt{50} \), we use:

Identify the factors:
  • \(50 = 2 \times 25\)
  • Then, break down 25: \(25 = 5 \times 5\)
  • Therefore, \(50 = 2 \times 5 \times 5\) = \(2 \times 5^2\)

  • Apply these factors through the square root: \(\sqrt{2 \cdot 5^2}\)

    • The square root of a prime square, \( \sqrt{5^2} \), is just the base prime, which is 5. Thus:
    • \(\sqrt{50} = 5\sqrt{2}\)
    This approach simplifies the radicand effectively, and the final result is \( 30 \sqrt{2} \) after multiplying by the coefficient.

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