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Chapter 7: Problem 34

Simplify the products. Give exact answers. $$\sqrt{5} \cdot \sqrt{7}$$

Short Answer

Expert verified
\(\sqrt{35}\)

Step by step solution

01

Identify the Property to Use

To simplify the product of two square roots, use the property of square roots which states that \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\).
02

Multiply the Radicands

Using the property identified, multiply the numbers inside the square roots: \(\sqrt{5} \times \sqrt{7} = \sqrt{5 \times 7}\).
03

Simplify the Product

Now multiply 5 and 7: \(\sqrt{5 \times 7} = \sqrt{35}\).

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Key Concepts

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

Square Roots
A square root asks what number, when multiplied by itself, equals the given number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9. We write this as \(\sqrt{9} = 3\ \).
Square roots are simplified to make math easier. Let's simplify the expression: \( \sqrt{5} \cdot \sqrt{7}\).
To solve problems with square roots, we need to understand key concepts like radicands and the multiplication property of square roots.
Radicands
The radicand is the number under the square root symbol. For instance, in the expression \(\sqrt{5} \), 5 is the radicand. The radicands in our exercise are 5 and 7.
When simplifying square roots, it is important to know how to work with radicands.
In the expression \(\sqrt{5} \cdot \sqrt{7}\):
  • Identify the radicands: 5 and 7.
  • Multiply the radicands: \(\sqrt{5 \cdot 7} = \sqrt{35}\).
Now you understand how radicands play a role in your square root calculations!
Multiplication Property of Square Roots
The multiplication property of square roots helps us simplify expressions involving square roots. It says that the product of two square roots is the square root of the product of the radicands. Written mathematically: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
This property is very useful when simplifying expressions like \(\sqrt{5} \cdot \sqrt{7}\):
  • First apply the property: \(\sqrt{5 \cdot 7} = \sqrt{35}\).
  • Then simplify the resulting square root if possible. In this case, since 35 cannot be factored into smaller square roots, \(\sqrt{35}\) is the final answer.
Now you can confidently use the multiplication property of square roots in your homework!

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Most popular questions from this chapter

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