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Chapter 1: Problem 5

Simplify each expression. $$\sqrt{256}$$

Short Answer

Expert verified
The simplified form of \(\sqrt{256}\) is 16.

Step by step solution

01

Identify the Expression to Simplify

We are tasked with simplifying the expression \(\sqrt{256}\). This means we need to find a number that, when multiplied by itself, equals 256.
02

Recognize the Type of Expression

The given expression involves a square root. To simplify it, we must find the square root of 256, which is a perfect square.
03

Find the Square Root

The number 256 is a perfect square. The square root of 256 is the number that multiplies by itself to yield 256. Calculate \(16 \times 16 = 256\). Therefore, the square root of 256 is 16.

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Key Concepts

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

Perfect Squares
In mathematics, a perfect square is a number that can be expressed as the product of an integer with itself. Essentially, if you can take an integer and multiply it by itself to arrive at a given number, that number is a perfect square.
Understanding perfect squares is crucial because they simplify the process of finding square roots.
  • Examples of perfect squares include 1, 4, 9, 16, 25, 36, and so on.
  • These numbers stem from the series of integers: 1, 2, 3, 4, 5, and 6.
Knowing the sequence of perfect squares up to a certain point can help you recognize them quickly in different mathematical problems. For instance, knowing that 256 is the product of 16 times 16 tells you that it is a perfect square.
Square Root Calculation
Calculating the square root involves finding a number that can be multiplied by itself to reach the desired value. For perfect squares, this process usually involves identifying the integer that achieves this.
For example, the square root of 256 requires you to think of which number, when squared, would give 256.
  • In this instance, the solution is 16, because \[16 \times 16 = 256\]
  • This method is generally straightforward when working with smaller perfect squares.
If a calculator is not available, remembering common perfect squares helps achieve efficiency in solving such problems. This comprehension of direct calculations is part of simplifying expressions related to squaring and square roots.
Mathematical Expressions Simplification
Simplifying mathematical expressions can sometimes be demanding but is made easier with awareness of rules and properties of numbers. Specifically for problems involving square roots and perfect squares, the process becomes systematic.
To simplify \(\sqrt{256}\):
  • Recognize that 256 is a perfect square.
  • Find the integer whose square equals 256; in this case, it is 16.
  • This reduces the expression \(\sqrt{256}\) to just 16.
The aim of simplification is to reduce an expression to its simplest form, eliminating any unnecessary complexity. Understanding these fundamental principles not only aids in solving such problems but also enhances your overall mathematical proficiency.

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