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Chapter 8: Problem 65

In Exercises \(55-68,\) multiply and, if possible, simplify. $$\sqrt{7 x} \cdot \sqrt{3 y}$$

Short Answer

Expert verified
\(\sqrt{21xy}\)

Step by step solution

01

Write down the problem

The first step is to write down the given problem: \(\sqrt{7 x} \cdot \sqrt{3 y}\)
02

Apply the multiplication rule for square roots

The multiplication rule for square roots states that the product of two square roots can be written as the square root of the product of the radicands, or numbers under the root. Therefore, \(\sqrt{7 x} \cdot \sqrt{3 y}\) can be written as \(\sqrt{(7x)*(3y)}\).
03

Compute the result under the square root

Perform the multiplication under the root to simplify: \(\sqrt{(7x)*(3y)} = \sqrt{21xy}\).

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Key Concepts

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

Simplifying Radicals
Radicals can often seem complicated, but the process of simplifying them can help in making calculations more straightforward. Simplifying a radical involves reducing it to its simplest form without changing its value.
To simplify a radical, look for factors that are perfect squares. For example, when simplifying \(\sqrt{12}\), recognize that 12 can be broken down into \(4\times3\). Since 4 is a perfect square (\(4 = 2^2\)), \(\sqrt{12}\) simplifies to \(2\sqrt{3}\).
  • Find the largest perfect square factor of the number under the radical.
  • Rewrite the radical as a product of two square roots if possible, with one being the perfect square.
  • Simplify the perfect square portion to its integer value.
This method reduces the expression while maintaining its original value. For algebraic expressions, factor out any terms completely encompassed by a square and simplify.
Multiplication Rule for Square Roots
The multiplication rule for square roots greatly simplifies expressions involving multiple radicals. This rule states that the product of two square roots is equal to the square root of the product of their radicands.
In mathematical terms, if you have two square roots \(\sqrt{a}\) and \(\sqrt{b}\), you can use the rule to write it as \(\sqrt{a}\times\sqrt{b} = \sqrt{ab}\).
  • Combine the radicands by multiplying them together.
  • Take the square root of this new product if possible.
Applying this rule helps in simplifying complex expressions effectively. For example, \(\sqrt{2}\times\sqrt{8} = \sqrt{16} = 4\), obtained by first calculating the product under the root and then computing the simplified square root.
Simplifying Expressions with Radicals
Simplifying expressions that contain radicals often involves several steps including applying multiplication rules and simplifying the resulting radical. It's useful to systematically approach these expressions to make them as simple as possible.
For example, given \(\sqrt{7x}\times\sqrt{3y}\), the following steps can be taken:
  • Use the multiplication rule for radicals to combine under a single radical: \(\sqrt{7x}\times\sqrt{3y} = \sqrt{21xy}\).
  • Look for any perfect square factors within the new radicand to further simplify.
  • If none, the expression \(\sqrt{21xy}\) is considered in its simplest form.
Simplifying expressions might not always lead to an integer solution, but they will reduce the complexity and make further operations more manageable. Always keep an eye out for factors that can be simplified and apply these principles skillfully for best results.

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Most popular questions from this chapter

Explain why \(a^{\bar{n}}\) is negative when \(n\) is odd and \(a\) is negative. What happens if \(n\) is even and \(a\) is negative? Why?

In Exercises \(75-82\), rationalize each denominator. Simplify, if possible $$\frac{2 x+4-2 h}{\sqrt{x+2-h}}$$

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$\left(\frac{1}{9}\right)^{-\frac{1}{2}}$$

Rationalize the denominator: \(\frac{1}{\sqrt[3]{2}}\)

When a radical expression has its denominator rationalized, we change the denominator so that it no longer contains a radical. Doesn't this change the value of the radical expression? Explain.
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