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Chapter 8: Problem 52

Find each product and simplify. $$ 5 \sqrt{6} \cdot 2 \sqrt{10} $$

Short Answer

Expert verified
20 \sqrt{15}

Step by step solution

01

Multiply the Coefficients

First, multiply the coefficients of the two square roots. The coefficients are 5 and 2.\(5 \times 2 = 10\)
02

Multiply the Radicals

Next, multiply the numbers inside the square roots. The radicals are \(\sqrt{6}\) and \(\sqrt{10}\).\(\sqrt{6} \times \sqrt{10} = \sqrt{6 \times 10} = \sqrt{60}\)
03

Simplify the Radical

Simplify \(\sqrt{60}\) by factoring out the largest perfect square. Here, \(60 = 4 \times 15\), and \(\sqrt{4} = 2\).\(\sqrt{60} = \sqrt{4 \times 15} = \sqrt{4} \times \sqrt{15} = 2 \sqrt{15}\)
04

Combine the Results

Finally, combine the results from Step 1 and Step 3:\(10 \times 2 \sqrt{15} = 20 \sqrt{15}\)

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Key Concepts

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

Multiplying Coefficients
Before diving into the radicals, we need to handle the coefficients.
Coefficients are the numbers in front of the square roots.
For the expression 5\sqrt{6} \, \cdot 2\sqrt{10}, the coefficients are 5 and 2.
You simply multiply 5 by 2, which equals 10.
So, the first step is:
  • Identify the coefficients
  • Multiply them
We get:\[5 \times 2 = 10\]This deals with the coefficients only.
Next, we move on to the radicals.
Multiplying Radicals
Now let's focus on the radicals, the parts under the square root symbols.
We have \sqrt{6} and \sqrt{10}. When multiplying radicals, combine the numbers under one square root.This means\[\sqrt{6} \times \sqrt{10} = \sqrt{6 \times 10} = \sqrt{60}\]
  • Multiply what's inside the square roots together
  • Combine them under a single radical
We're left with \sqrt{60}. However, our work isn鈥檛 finished yet鈥攊t's time to simplify it.
Simplifying Radicals
Simplifying radicals involves breaking them down into easier components. Here, \sqrt{60} can be simplified by factoring out the largest perfect square.
A perfect square is a number like 1, 4, 9, 16, that can be square rooted cleanly.The largest perfect square factor of 60 is 4 because:
  • 60 = 4 \times 15
  • \sqrt{60} = \sqrt{4 \times 15}
Since \sqrt{4} = 2, we can rewrite this as:\[\sqrt{60} = \sqrt{4} \times \sqrt{15} = 2 \sqrt{15}\]Now we have our simplified radical.
Factoring Perfect Squares
Factoring perfect squares is key to simplifying radicals.
You look for numbers like 4, 9, or 16 within the radical.
For example, in \sqrt{60}, the largest perfect square factor is 4.
  • Factor 60 into 4 \times 15
  • Simplify \sqrt{4} to 2
Then, rewrite the radicals with the perfect square factored out:\[\sqrt{60} = 2 \sqrt{15}\]Once we have the simplified form, we can then combine results from all steps.

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Most popular questions from this chapter

Simplify each radical expression. $$ \sqrt{\frac{7}{16}} $$

Simplify each radical expression. $$ \sqrt{\frac{13}{25}} $$

To rationalize the denominator of an expression such as \(\frac{4}{\sqrt{3}},\) we multiply both the numerator and denominator by \(\sqrt{3}\). By what number are we actually multiplying the given expression, and what property of real numbers justifies the fact that our result is equal to the given expression?

Solve each equation. (Hint: In Exercises 67 and 68, extend the concepts to fourth root radicals.) $$ \sqrt[3]{3 x^{2}-9 x+8}=\sqrt[3]{x} $$

Find each product and simplify. Simplify the product \(\sqrt{8} \cdot \sqrt{32}\) in two ways. First, multiply 8 by 32 and simplify the square root of this product. Second, simplify \(\sqrt{8},\) simplify \(\sqrt{32, \text { and then multiply. }}\) How do the answers compare? Make a conjecture (an educated guess) about whether the correct answer can always be obtained using either method when simplifying a product such as this.
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