Problem 57 - Algebra A Combined Function

Question

Simplify. Assume that the variables represent any real number. $$ \sqrt{(x-5)^{2}} $$

Step-by-Step Solution

Answer

The simplified form is \(|x-5|\).

Step 1: Identify the Expression Inside the Square Root

The expression given in the problem is \( \sqrt{(x-5)^{2}} \). We need to simplify this by considering what's under the square root sign.

Step 2: Understand Removing the Square Root and Square Function

Since we're dealing with the square root of a square, \( \sqrt{a^2} = |a| \) for any real number \( a \). This means \( \sqrt{(x-5)^{2}} = |x-5| \).

Step 3: Final Simplification

The expression simplifies to \( |x-5| \). This is because the square root and square cancel out, leaving the absolute value of the expression \( x-5 \).

Key Concepts

Absolute Value

When we talk about absolute value, we are referring to the distance of a number from zero on the number line, regardless of direction. It is always a non-negative number.
In simple terms, the absolute value of a number is its 'magnitude' without considering its sign. For example, the absolute value of both \( 7\) and \( -7\) is \( 7\), because they are equally distant from zero.
In the context of our given problem, \( |x-5|\) represents the absolute value of \( x-5\), meaning it is the distance of \( (x-5)\) from zero, no matter if \( (x-5)\) is positive or negative.

• The absolute value function \(|a|\) ensures that the outcomes are always non-negative.
• It is used widely in problems where direction (positive or negative) doesn't matter, only the magnitude does.

Square Root Properties

Understanding the properties of square roots is key to simplifying expressions like \(\sqrt{(x-5)^{2}}\).
One fundamental property is that a square root and a square cancel each other out. However, to correctly interpret this, we must incorporate absolute value.
So, for any real number \(a\), the property \(\sqrt{a^2} = |a|\) holds true. This property ensures that the result is always a non-negative number, reflecting the true distance (absolute value) of \(a\) from zero.

• Square roots of perfect squares lead to absolute values, due to the need to always represent distance positively.
• This avoids any ambiguity related to positive or negative results, harmonizing with the concept of absolute value.

Real Numbers

Real numbers include all the numbers you can think of on the number line - both rational and irrational numbers. These numbers make up everything from fractions and integers to decimals and infinite non-repeating decimal numbers.
In our exercise, we assumed that the variable \(x\) represents any real number when simplifying \(\sqrt{(x-5)^{2}}\). This is important because real numbers are the typical domain when dealing with square roots and absolute values.
Understanding real numbers means recognizing:

• Rational numbers such as integers and fractions.
• Irrational numbers like \(\sqrt{2}\) or \(rac{\pi}{3}\) which cannot be expressed as fractions.
• All these collectively sit on the continuum of the number line.

Real numbers are crucial in mathematics, giving context to operations like taking square roots and ensuring that expressions are valid across a broad range of values.