Problem 67 - Algebra 2

Question

Simplify. $$ \sqrt[4]{(x-5)^{2}} $$

Step-by-Step Solution

Answer

\( \sqrt{x-5} \)

Step 1: Identify the Problem

We are asked to simplify the expression \( \sqrt[4]{(x-5)^{2}} \). This involves simplification of a radical expression which is in the form of the fourth root of something squared.

Step 2: Use Radical and Exponent Rules

A key step in simplification is to recognize that the fourth root can be rewritten using exponents as \( (x-5)^{2/4} \) because taking the fourth root is the same as raising to the power \( \frac{1}{4} \).

Step 3: Simplify the Exponent

Simplify the exponent \( \frac{2}{4} \), which simplifies to \( \frac{1}{2} \). This is because \( \frac{2}{4} = \frac{1}{2} \), reducing the expression to \( (x-5)^{1/2} \).

Step 4: Convert Back to Radical Form

Convert the simplified exponent back to a radical, which gives \( \sqrt{x-5} \), as raising to the power of \( \frac{1}{2} \) is equivalent to taking the square root.

Key Concepts

Fourth Root

When you see an expression like \( \sqrt[4]{x} \), this is known as the fourth root. Here, the number 4 is referred to as the index of the root. Taking the fourth root of a number or expression is like asking, 'what number times itself four times equals the given number?' For example, if \( 16 \) is the number, the fourth root of \( 16 \) is \( 2 \) because \( 2 \times 2 \times 2 \times 2 = 16 \). This means the fourth root helps you break down numbers or expressions into factors that multiply together four times to get the original number.

If you have an expression like \( \sqrt[4]{(x-5)^2} \), you can see that we are dealing with a number or expression raised to a power, and then taking the fourth root. Using radical notations such as this one can sometimes be tricky, but it allows the expression to be rewritten in exponential terms, which can make simplification easier. Understanding and recognizing the fourth root allows you to see the relationship between powers and roots more clearly.

Simplifying Exponents

Simplifying exponents is about making an expression more straightforward by reducing the power terms. When you have an expression such as \( (x-5)^{2/4} \), you are dealing with exponents. To simplify this, you should look at the fraction \( \frac{2}{4} \), which can be reduced to \( \frac{1}{2} \).

Why does this work? When you divide the numerator by the denominator, you simplify the exponent to its easiest form. This shows the importance of breaking down fractions from a larger form into the simplest form possible. After simplification, the expression \( (x-5)^{2/4} \) becomes \( (x-5)^{1/2} \).

Remember, simplifying exponents is a crucial skill, as it aids in making complex expressions more manageable and easier to understand or solve.

Square Root

The square root is one of the most common radical expressions you'll encounter. Typically denoted as \( \sqrt{x} \), it represents a value that, when multiplied by itself, will give the original number \( x \). For example, the square root of \( 9 \) is \( 3 \) because \( 3 \times 3 = 9 \).

When you see an example like \( \sqrt{x-5} \), it indicates finding a number that when squared, results in \( x-5 \). In our context, the transformation from \( (x-5)^{1/2} \) to \( \sqrt{x-5} \) shows how rational exponents transform into square root expression. Being comfortable with switch between these forms involves recognizing that raising something to the power \( \frac{1}{2} \) is synonymous with taking the square root, making it an invaluable tool in simplifying mathematical expressions.

Rational Exponents

Rational exponents provide a link between radical notation and exponential notation. Instead of writing a root using radical signs, it's possible to express it with a power or exponent that is a fraction. For instance, the fourth root of an expression can be rewritten using rational exponents as \( (x-5)^{1/4} \). Here, the exponent's numerator indicates the power, while the denominator indicates the root.

In our exercise, we initially had \( (x-5)^{2/4} \), where 2 is the power and 4 is the root. By simplifying this to \( (x-5)^{1/2} \), we converted it into a form that can further be expressed in radical terms as a square root, \( \sqrt{x-5} \).

Understanding rational exponents is particularly useful for elegantly solving and simplifying expressions and equations, converting between different formats, and increasing math fluency.