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Chapter 10: Problem 39

Evaluate or simplify each expression. (Assume \(x \geq 0 .)\) $$\sqrt{25}-\sqrt{9}$$

Short Answer

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Step by step solution

01

Evaluate the square root of 25

Find the square root of 25. Since 25 is a perfect square, \(\text{the square root of 25 is } 5 \).
02

Evaluate the square root of 9

Find the square root of 9. Since 9 is a perfect square, \(\text{the square root of 9 is } 3 \).
03

Subtract the square roots

Subtract the square root of 9 from the square root of 25: \(\text{5 - 3 = 2.}\)

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Key Concepts

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

perfect squares
Understanding perfect squares is essential for evaluating square roots. A perfect square is a number that is the square of an integer. In other words, it is the result of an integer multiplied by itself. For example:
  • The number 25 is a perfect square, because it can be expressed as \(5^2\) (5 times 5).
  • The number 9 is also a perfect square, as it can be written as \(3^2\) (3 times 3).
Recognizing perfect squares helps simplify square roots because knowing them allows you to quickly identify and evaluate these roots. This concept is the foundation for the exercise provided.
square root evaluation
Evaluating square roots involves finding a number that, when multiplied by itself, equals the original number. For instance, for the given problem:

Step 1:
The square root of 25: Since 25 is a perfect square, its square root is \(\sqrt{25} = 5\).

Step 2:
The square root of 9: Similarly, 9 is a perfect square, and its square root is \(\sqrt{9} = 3\).

By recognizing and understanding perfect squares, it becomes easier to evaluate such square roots. It is crucial to memorize basic perfect squares such as 1, 4, 9, 16, 25, 36, etc., as this will make problems like these much more approachable.
basic arithmetic operations
Basic arithmetic operations are fundamental in mathematics. In this context, we will focus on subtraction, which is used in the final step of the exercise:

  • Subtraction: This operation involves taking one number away from another. For instance, in the given exercise, after evaluating the square roots, we subtract the results.
  • Example: The square root of 25 is 5, and the square root of 9 is 3. Subtract these results: \(5 - 3 = 2\).
It鈥檚 important to remember the order of operations to ensure accurate results. Practice these steps with various numbers to gain more confidence.

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Most popular questions from this chapter

Sketch the graph of each equation in a rectangular coordinate system. Label the intercepts. $$x=2$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{25}+\sqrt{24}$$

Rationalize the denominators and simplify. $$(4-\sqrt{7})^{2}-\frac{18}{4+\sqrt{7}}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{5}+\sqrt{\frac{1}{5}}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$6 \sqrt{8}-\sqrt{98}$$
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