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Chapter 7: Problem 1

Use the product rule to multiply. See Example \(I\). \(\sqrt{7} \cdot \sqrt{2}\)

Short Answer

Expert verified
The product of \( \sqrt{7} \) and \( \sqrt{2} \) is \( \sqrt{14} \).

Step by step solution

01

Identify the Operation

The given expression is \( \sqrt{7} \cdot \sqrt{2} \). We are required to multiply these two square roots using the product rule.
02

Apply the Product Rule for Radicals

The product rule for radicals states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). We apply this rule to our expression: \( \sqrt{7} \cdot \sqrt{2} = \sqrt{7 \cdot 2} \).
03

Simplify the Expression

Calculate the product inside the square root: \( 7 \cdot 2 = 14 \). Therefore, \( \sqrt{7 \cdot 2} = \sqrt{14} \). This is the simplified form of the product of the two radicals.

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Key Concepts

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

Multiplying Radicals
When you encounter a problem that involves multiplying radicals, the product rule for radicals becomes your best friend. This rule simplifies the process by allowing you to combine two radical expressions under a single square root.
  • According to the product rule: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
  • This means you can multiply the numbers inside the radicals together, then take the square root of the result.
By using this approach, you simplify the expression step by step. Remember, the multiplied result inside the radical should never include negative numbers since the square root of a negative is not defined in the set of real numbers.
Simplifying Square Roots
After applying the product rule to multiply two radicals, your next task is often to simplify the square root that results from your multiplication.
  • For example, if you end up with \( \sqrt{14} \), check whether the calculator value can be reduced further.
  • Think of if you can break down any part of 14 into perfect squares, like \( \sqrt{4} \cdot \sqrt{2} \). Since 14 doesn't break down further, it stays as \( \sqrt{14} \).
Simplifying is all about finding common factors or just recognizing, in some cases, that the number inside the radical is as simple as it can be. Practicing simplification helps you understand when a radical is in its most basic form.
Radical Expressions
Radical expressions involve roots, such as square roots, cube roots, and so on. These expressions can often appear intimidating, but once you understand the rules, they become much more manageable.
  • At the core, a radical looks like \( \sqrt{x} \), which signifies the square root of \( x \).
  • Every radical expression follows specific rules, like the product rule, that guide how we can simplify or manipulate them.
Knowing how to apply these rules is essential in math. Radical expressions can be simplified, multiplied, and even added or subtracted, providing flexibility in how they are used in equations. By grasping these basic principles, you become more adept at handling any problems involving radicals.

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Most popular questions from this chapter

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[6]{4} $$

Rationalize each denominator. See Example 4. $$ \frac{6}{2-\sqrt{7}} $$

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Use a calculator to approximate each square root to 3 decimal places. $$ \sqrt{56} $$

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