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Chapter 9: Problem 10

For Problems \(1-20\), multiply and simplify where possible. $$ \sqrt{12} \sqrt{20} $$

Short Answer

Expert verified
The simplified form is \( 4\sqrt{15} \).

Step by step solution

01

Multiply the Radicals

Multiply the expressions inside the square roots. \[ \sqrt{12} \times \sqrt{20} = \sqrt{12 \times 20} \]
02

Compute the Product Inside the Radical

Calculate the product of 12 and 20.\[ 12 \times 20 = 240 \] Thus, \[ \sqrt{12 \times 20} = \sqrt{240}. \]
03

Simplify the Square Root

Break down 240 into its prime factors to simplify.\[ 240 = 2^4 \times 3 \times 5 \]Now, use the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to simplify:\[ \sqrt{240} = \sqrt{2^4 \times 3 \times 5} = \sqrt{2^4} \times \sqrt{3} \times \sqrt{5} \] Since \(\sqrt{2^4} = 4\), it becomes:\[ 4 \times \sqrt{3} \times \sqrt{5} \] Combine:\[ 4 \sqrt{15} \]
04

Conclusion

The simplified form of \( \sqrt{12} \times \sqrt{20} \) is \( 4 \sqrt{15} \).

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Key Concepts

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

Prime Factorization
Prime factorization is a technique used to express a number as the product of its prime numbers. These are numbers greater than 1 that have no divisors other than 1 and themselves. For instance, to factorize 240, which we encounter in our problem, we break it down step-by-step into smaller components:
  • Start with the smallest prime, 2. Divide 240 by 2 to get 120.
  • Divide 120 by 2 again to get 60. Continue this process until you can no longer divide by 2 evenly, resulting in 2.
  • Next, move to the prime number 3 and divide the remaining product (15) by 3 to get 5, which is also a prime number.
Markdown representation: \(240 = 2^4 \times 3 \times 5\)
This method is crucial for simplifying square roots as it allows us to break down the number inside the radical, revealing any perfect squares that can be extracted.
Multiplying Radicals
Multiplying radicals can seem tricky at first, but it obeys the same basic rules of multiplication. When you multiply two square roots, you multiply the numbers inside the roots first, as expressed in the original exercise. For our exercise:
  • The square roots \(\sqrt{12}\) and \(\sqrt{20}\) multiply to become \(\sqrt{12 \times 20}\).
  • The product equals \(\sqrt{240}\).
This step simplifies the problem by combining it into a single square root, which can then be simplified using other mathematical properties. Always remember, under the same index, simply multiply the numbers beneath the radicals and then simplify.
Square Root Properties
Simplifying square roots relies heavily on its properties, especially when dealing with multiplication.
  • The most essential property used here is: \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\). This means a product inside a square root can be separated into the product of two square roots.
  • In our solution, this property is used with \(\sqrt{240}\) to break it down into its components: \(\sqrt{2^4} \cdot \sqrt{3} \cdot \sqrt{5}\).
  • Since \(\sqrt{2^4} = 4\), you extract 4 from under the root, reducing the expression to \(4 \sqrt{15}\).
Understanding and applying this property allows simplification by revealing and removing perfect squares, leaving the simplest form of radicals.

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Most popular questions from this chapter

For Problems \(55-70\), rationalize the denominators and simplify. All variables represent positive real numbers. $$ \frac{2+\sqrt{3}}{3-\sqrt{2}} $$

Use a calculator to evaluate each radical. (Objective 1) $$\sqrt{784}$$

Evaluate each radical without using a calculator or a table. (Objective 1) $$\sqrt{0.0121}$$

Evaluate each radical without using a calculator or a table. (Objective 1) $$\sqrt[3]{64}$$

Find a rational approximation, to the nearest tenth, for each radical expression. (Objective 2) $$9 \sqrt{3}+\sqrt{3}$$
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