Problem 5 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\sqrt{256}\)

Step-by-Step Solution

Answer

The simplified expression is 16.

Step 1: Identify the Expression

The expression given is \( \sqrt{256} \). This expression asks us to find the square root of 256.

Step 2: Determine the Square Root

The square root of a number \( x \) is a number \( y \) such that \( y^2 = x \). We need to find a number \( y \) such that \( y^2 = 256 \).

Step 3: Check Perfect Squares

Recognize that 256 is a perfect square. A perfect square is a number that is the square of an integer. Check integers: \( 16^2 = 256 \), so \( 16 \) is the square root of 256.

Step 4: Simplify the Expression

Since we found that \( 16^2 = 256 \), we simplify \( \sqrt{256} \) as \( 16 \). This is the simplest form of the expression.

Key Concepts

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. It's like finding a number's twin that, when paired together, completes the expression. In mathematical terms, if you square a number 'y', it should result in 'x'. This is expressed as \( y^2 = x \).
For example, the square root of 9 is 3, because \( 3^2 = 9 \). Similarly, finding the square root of 256 involves identifying a number that squares to give 256.

• Symbolically represented by \( \sqrt{ } \).
• Always non-negative in basic arithmetic.

In our original exercise, calculating \( \sqrt{256} \) yields 16, since \( 16 \times 16 = 256 \). Always remember, for every positive number, there is both a positive and a negative square root, but we usually take the positive root.

Perfect Squares

Perfect squares are numbers that can be written as the product of an integer multiplied by itself. In other words, these are the results of squaring whole numbers. Recognizing perfect squares helps simplify square root expressions quickly.

• Examples of perfect squares include: 1 (\(1^2\)), 4 (\(2^2\)), 9 (\(3^2\)), 16 (\(4^2\)), and so forth.
• These numbers form a sequence: 1, 4, 9, 16, 25, 36, 49, etc.
• Perfect squares are like the steps on a ladder - each step perfectly balanced and clear.

In our case with \( \sqrt{256} \), recognizing 256 as a perfect square helps us directly identify that \( 16^2 = 256 \), simplifying our task.

Algebra Problem Solving

Algebra is more than numbers and symbols; it's about finding the unknown and simplifying expressions. When tasked with simplifying algebraic expressions involving square roots, you'll often look for patterns or known quantities.
Here are practical steps to approach such problems:

• Identify the nature of the expression - Is it a simple root, quadratic, or something else?
• Look for perfect squares to simplify expressions immediately.
• Structure your solution - Write down what you're tasked to find and any patterns noted during calculation.

In algebra, it’s essential to break down the problem systematically, as seen with the solution to \( \sqrt{256} \). Identifying that 256 is a perfect square allowed for quick problem solving, demonstrating algebra's power to simplify complexity into a clear solution.