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Chapter 1: Problem 22

Evaluate each radical. $$\sqrt{25-9}$$

Short Answer

Expert verified
The answer is 4.

Step by step solution

01

- Simplify the expression inside the radical

First, simplify the expression inside the radical sign. The expression given is \(25 - 9\). Perform the subtraction: \(25 - 9 = 16\).
02

- Evaluate the square root

Now, take the square root of the simplified expression. The square root of 16 is 4, because \(4 \times 4 = 16\). Therefore, \(\sqrt{16} = 4\).

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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 an essential concept in algebra. They allow you to find a number which, when multiplied by itself, gives the original number. For example, the square root of 16 is a number that, when squared, equals 16. That number is 4 because 4 times 4 equals 16.

In mathematical notation, the square root of a number x is written as \( \sqrt{x} \). So, \( \sqrt{16} = 4 \).

Key points to remember when evaluating square roots:
  • Negative numbers do not have real square roots since a square is always positive or zero.
  • Always check if the number inside the square root is a perfect square for easier calculation.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. This step is crucial when dealing with radicals. Before taking the square root, it's often helpful to simplify the expression inside the radical sign.

For example, consider the expression \( \sqrt{25-9} \). First, simplify the expression inside the radical:
  • Perform any subtraction, addition, multiplication, or division inside the radical first. Here, calculate \( 25 - 9 = 16 \).
  • Then, you can take the square root of the simplified expression.

Simplifying helps you manage the complexity of the expressions and makes it easier to perform other algebraic operations.
Step-by-Step Problem Solving
Solving problems step-by-step ensures accuracy and clarity. Here's how to evaluate a radical like \( \sqrt{25-9} \), step-by-step:

Step 1: Simplify the expression inside the radical. In \( \sqrt{25-9} \), first simplify inside the radical sign, so:
  • Calculate \( 25 - 9 \).
  • This gives \( 16 \).

Step 2: Evaluate the square root. Now that we have simplified the expression to \( \sqrt{16} \):
  • Find the square root of \( 16 \).
  • The square root of 16 is 4, since \( 4 \times 4 = 16 \).

This step-by-step approach ensures you won't miss any important details and helps you understand each part of the problem. Keep practicing to get better!

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