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Chapter 10: Problem 20

Evaluate or simplify each expression. (Assume \(x \geq 0 .)\) $$\sqrt{324}$$

Short Answer

Expert verified
The square root of 324 is 18.

Step by step solution

01

Understand the Problem

The task is to evaluate the square root of 324. This means finding a number which, when squared, gives 324.
02

Identify Perfect Squares

Recall perfect squares: 1, 4, 9, 16, 25, 36, ..., 324. Recognizing that 324 is a perfect square is crucial.
03

Use Knowledge of Perfect Squares

From the list of perfect squares, identify that 18 is the number such that 18 squared is 324. This means \(18^2 = 324\).
04

Evaluate the Square Root

Since \((18)^2 = 324\), it follows that \(\root{(324)} = 18\).

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Key Concepts

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

Perfect Squares
A perfect square is a number that can be expressed as the product of an integer with itself. For example, 1, 4, 9, 16, 25, and so on, are all perfect squares because they are the squares of 1, 2, 3, 4, 5, respectively. Understanding perfect squares is essential when evaluating or simplifying square roots.
Recognizing that 324 is a perfect square helps significantly in our problem. We see that \(18^2 = 324\), so \(18\) is the square root of 324. The ability to identify perfect squares allows us to simplify the process of finding square roots effectively.
Simplifying Expressions
Simplifying expressions involves reducing them to their most straightforward form. In our exercise, we simplify \(\sqrt{324}\) by recognizing 324 as a perfect square. The simplified form of \(\sqrt{324}\) is \(18\) because \(18^2 = 324\).
To simplify a square root, follow these steps:
  • Determine if the number is a perfect square.
  • If it is, write the square root as the integer whose square equals the original number.
  • This approach minimizes complex calculations and reduces mistakes.
By simplifying expressions, we make them easier to work with, and results become clearer and more understandable.
Algebraic Operations
Algebraic operations include addition, subtraction, multiplication, division, and finding roots of numbers. These operations are fundamental in solving various mathematical problems. In our exercise, we specifically deal with finding the square root, which is an algebraic operation.
To find the square root, follow these steps:
  • Identify if the number is a perfect square.
  • Use the knowledge of perfect squares to find the root.
  • Write the simplified form of the expression.
Understanding algebraic operations gives us the tools to manipulate and solve mathematical problems confidently. In this case, knowing that \(18^2 = 324\) allows us to determine \(\sqrt{324} = 18\).

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Most popular questions from this chapter

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. $$4 \sqrt{12}-\sqrt{75}$$

Rationalize the denominators and simplify. $$\frac{6}{3-\sqrt{7}}-\frac{21}{\sqrt{7}}$$

Verify that \(2+\sqrt{10}\) is a solution to the equation \(x^{2}-4 x-6=0\).

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. $$\sqrt{12}+\sqrt{27}$$

Sketch the graph of each equation in a rectangular coordinate system. Label the intercepts. $$y+2=0$$
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