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Chapter 10: Problem 1

Use the product rule to multiply. $$\sqrt{3} \cdot \sqrt{5}$$

Short Answer

Expert verified
The multiplication of \( \sqrt{3} \) and \( \sqrt{5} \) is \(\sqrt{15}\)

Step by step solution

01

Product Rule Application

Apply the product rule for square roots, that is, multiply the numbers under the square roots. In this case, multiply 3 by 5 which results in 15.
02

Simplify the expression

The square root of the product is the product of the square roots. Hence, the simplified form of the given expression is \( \sqrt{15} \).

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Key Concepts

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

Square Roots
Square roots are mathematical expressions that represent a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. Square roots are often denoted with the radical symbol (\( \sqrt{} \)).
When you encounter a square root, it asks for a value that, when squared, results in the number under the radical symbol. This process helps in various operations involving numbers and equations.
To compute square roots, especially in simple numbers, consider these points:
  • Even perfect squares, like 4, 9, 16, have whole numbers as their square roots (2, 3, 4 respectively).
  • For non-perfect squares, the square roots can be approximated as decimals.
  • Square roots are used extensively in algebra and geometry to simplify expressions and solve equations.
Multiplication of Radicals
Multiplying radicals involves the use of the product rule, which is a key principle to understand. The product rule for radicals states that the square root of a product is the product of the square roots. This means \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
Let's use an example to make this clearer:
  • Consider \( \sqrt{3} \cdot \sqrt{5} \). According to the product rule, you multiply the numbers under the radicals: 3 and 5.
  • Multiply these numbers to get 15, so the expression becomes \( \sqrt{15} \).
  • This rule simplifies the multiplication process of radicals significantly, allowing you to combine them into a single radical expression.
Understanding and applying the product rule for radicals makes complex problems far more manageable and is an essential skill in algebra.
Simplifying Radicals
Simplifying radicals involves writing a radical expression in its simplest form. This often means ensuring there are no perfect square factors inside the radical symbol, other than 1.
To simplify a radical:
  • First, determine if the number under the radical sign can be divided by a perfect square.
  • If divisible, factor out the perfect square, simplifying this part outside the radical.
  • If no perfect squares are found in the factors, the radical is already in its simplest form.
For example, given \( \sqrt{15} \), notice that 15 has prime factors 3 and 5. Since neither is a perfect square, \( \sqrt{15} \) is already simplified. Simplifying radicals is a useful process, especially in solving equations and breaking down complex expressions into more manageable forms.

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Most popular questions from this chapter

In Exercises \(93-104\), rationalize each numerator. Simplify, if possible. $$\frac{\sqrt{x+5}-\sqrt{x}}{5}$$

In Exercises \(39-64,\) rationalize each denominator. $$\frac{3 x y^{2}}{\sqrt[5]{8 x y^{3}}}$$

Exercises \(147-149\) will help you prepare for the material covered in the first section of the next chapter. Solve by factoring: \(x^{2}=9\)

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{25}{5 \sqrt{2}-3 \sqrt{5}}$$

What are conjugates? Give an example with your explanation.
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