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Chapter 1: Problem 661

Simplify. $$ \sqrt{64} $$

Short Answer

Expert verified
The simplified form of \(\sqrt{64}\) is 8.

Step by step solution

01

Understand the Problem

The task is to simplify the expression \(\sqrt{64}\). This means finding the number that, when multiplied by itself, gives 64.
02

Identify the Perfect Square

Recognize that 64 is a perfect square. A perfect square is a number that can be expressed as the product of an integer with itself.
03

Determine Which Number Squared Equals 64

Find the integer \(n\) that satisfies \(n^2 = 64\). Consider that 8 \(times\) 8 equals 64.
04

Write the Simplified Form

Since \(8^2 = 64\), the simplified form of \(\sqrt{64}\) is 8.

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Key Concepts

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

Perfect Square
Let's start by understanding what a perfect square is. A perfect square is a number that can be written as a product of an integer by itself. For example, 16 is a perfect square because it can be written as 4 times 4 (or \(4^2\)). Similarly, 64 is a perfect square because it can be written as 8 times 8 (or \(8^2\)).

Square Root
Next, we need to talk about square roots. The square root of a number is a value that, when multiplied by itself, equals the original number. For instance, the square root of 25 is 5, because \(5 \times 5 = 25\).
Another example is the square root of 36, which is 6, because \(6 \times 6 = 36\).

To simplify a square root, like \(\sqrt{64}\), you need to find what number multiplied by itself equals the number under the root. \(\sqrt{64}\) simplifies to 8 because \(8 \times 8 = 64\).

Integer
Finally, let's clarify what an integer is. An integer is any whole number, positive or negative, including zero. Examples of integers are -3, 0, 1, and 7.
In our case, we identified that 8 is the square root of 64. It鈥檚 important to note that 8 is an integer.

So, when simplifying square roots, we always aim to express the result as an integer, when possible. This makes our calculations and understanding much simpler.

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