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Chapter 9: Problem 3

For Problems \(1-20\), multiply and simplify where possible. $$ \sqrt{6} \sqrt{8} $$

Short Answer

Expert verified
The expression simplifies to \( 4\sqrt{3} \).

Step by step solution

01

Multiply the Radicals

To multiply the radicals \( \sqrt{6} \) and \( \sqrt{8} \), we use the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). So, \( \sqrt{6} \cdot \sqrt{8} = \sqrt{6 \cdot 8} = \sqrt{48} \).
02

Simplify the Radical

The next step is to simplify \( \sqrt{48} \). We find the prime factorization of 48, which is \( 48 = 2^4 \times 3 \). Knowing \( \sqrt{48} = \sqrt{2^4 \times 3} = \sqrt{(2^2)^2 \times 3} = 2^2 \cdot \sqrt{3} \), simplify further: \( 2^2 = 4 \), so \( \sqrt{48} = 4\sqrt{3} \).

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Key Concepts

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

Simplifying Radicals
When we talk about simplifying radicals, we're trying to make the expression inside the square root as simple as possible. This often means reducing the size of the number or expression inside. In many cases, you might start with a product like \( \sqrt{48} \), a snapshot of what happens by multiplying \( \sqrt{6} \) and \( \sqrt{8} \). The golden rule here is to factor the number inside the square root thoroughly.
  • Look for perfect square factors: numbers that can be written as another number squared, like \( 4 = 2^2 \) or \( 9 = 3^2 \).
  • Rewrite the radical in terms of these perfect squares.
  • The goal is to "take out" these perfect squares out of the square root, because the square root of \( x^2 \) is simply \( x \).
By breaking down \( \sqrt{48} \) into \( \sqrt{16 \times 3} \) or \( 4\sqrt{3} \), we make simplifying larger numbers easier and manageable.
Prime Factorization
Prime factorization is the process of breaking down a number into its most basic building blocks: prime numbers. Prime numbers are numbers greater than 1 that only have two divisors, 1 and themselves. To simplify radicals, knowing the prime factorization is crucial because it allows us to find perfect squares.
  • Start by dividing the number by the smallest prime number, which is 2, until it no longer divides evenly.
  • Move to the next smallest prime number, like 3, and continue the process.
  • Write down each step to see the primes that multiply together to make the original number.
For example, with 48, through prime factorization, we find: \( 48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3 \). By understanding these factors, we easily identify that \( 2^4 \) can be split into \( (2^2)^2 \), which simplifies to \( 4 \) outside of the square root.
Properties of Square Roots
Understanding the fundamental properties of square roots helps simplify expressions effectively. Square roots have several useful properties that make working with them more straightforward. Two vital properties are multiplication and division:
  • Multiplication Property: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). This property allows combining square roots into a single expression, as we did: \( \sqrt{6} \cdot \sqrt{8} = \sqrt{48} \).
  • Division Property: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This helps divide numbers under a root.
These properties show how square roots simplify complicated expressions. When applied correctly, they maintain balance while simplifying the radicals by allowing easy multiplication or division of square root terms. Thus, they are essential rules to remember when dealing with radicals efficiently.

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Most popular questions from this chapter

Use a calculator to evaluate each radical. (Objective 1) $$\sqrt{2304}$$

Evaluate each radical without using a calculator or a table. (Objective 1) $$\sqrt{144}$$

Evaluate each radical without using a calculator or a table. (Objective 1) $$-\sqrt[3]{729}$$

Simplify each expression by using the distributive property. (Objective 2) $$8 \sqrt[3]{6}-10 \sqrt[3]{6}$$

Use a calculator to evaluate each radical. (Objective 1) $$\sqrt[3]{5832}$$
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