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Chapter 1: Problem 659

Simplify. $$ \sqrt{36} $$

Short Answer

Expert verified
The simplified form of \( \sqrt{36} \) is 6.

Step by step solution

01

Identify the Expression

The given expression is \( \sqrt{36} \). This is a square root expression.
02

Recall the Definition of Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{a} = b \) means \( b^2 = a \).
03

Find the Square Root of 36

Determine which number multiplied by itself equals 36. Since \( 6 \times 6 = 36 \), it follows that \( \sqrt{36} = 6 \).

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Key Concepts

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

Definition of Square Root
A square root is a fundamental concept in mathematics. It refers to a value that, when multiplied by itself, returns the initial number. For instance, if we have a number \(a\), its square root will be denoted as \(\sqrt{a}\). This means that if \(b\) is the square root of \(a\), then \(b^2 = a\). In our exercise, \(\sqrt{36}\), we find the square root of 36 by identifying a number which, when squared, results in 36.

Understanding this concept helps a lot because it simplifies complex expressions and makes solving equations easier. So, when you encounter a square root symbol, you are looking for a number that squares back to the given number.
Simplifying Square Roots
Simplifying square roots means expressing the square root in its simplest form. For instance, \(\sqrt{36}\) simplifies to 6, since 6 multiplied by 6 equals 36.

Let's break down the process:
  • Identify the number inside the square root.
  • Determine if the number is a perfect square (a number that can be expressed as the square of an integer).
  • If it is a perfect square, find the integer that, when squared, equals the original number.
Since 36 is a perfect square (\(6\times6\) = 36), the square root of 36 is 6.

This ability to simplify square roots is a very useful skill, particularly in more complicated mathematical problems.
Basic Arithmetic Operations
Basic arithmetic operations include addition, subtraction, multiplication, and division. When working with square roots, multiplication is particularly important.
For example, when simplifying \(\sqrt{36}\), we see that 6 multiplied by itself gives us 36, which is the basis of understanding square roots.

Here鈥檚 why these operations matter:
  • Multiplication: Helps in identifying the square root (e.g., \(6脳6 = 36\).
  • Division: Sometimes you need to divide numbers under the square root to simplify them.
When you know these basic operations well, it becomes easier to handle and simplify square roots and other mathematical expressions seamlessly.

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