Problem 73 - College Algebra Essentials

Question

Simplify the radical expressions in Exercises \(67-74\) if possible. $$\frac{\sqrt[5]{64 x^{6}}}{\sqrt[5]{2 x}}$$

Step-by-Step Solution

Answer

The simplified radical expression is \(2x\)

Step 1: Express the 5th root of each number

First, rewrite each cube root as a power: \[ \frac{(64 x^{6})^{1/5}}{(2 x)^{1/5}} \]

Step 2: Simplify the numbers and variables

We can simplify the numerators and denominators: \[ \frac{2^{6/5} x^{6/5}}{2^{1/5} x^{1/5}} \]

Step 3: Use exponent subtraction rule

When dividing like bases, subtract the exponents: \[ 2^{(6/5 - 1/5)} x^{(6/5 - 1/5)} \]

Step 4: Subtract the exponents

Subtracting the exponents respectively gives: \[ 2^{1} x^{1} \]

Key Concepts

Exponent Rules

Exponent rules are essential for simplifying expressions involving powers and roots. They help us to manipulate and reduce expressions in a systematic way. Let's explore a few exponent rules that are frequently used:

• Product Rule: When multiplying like bases, add the exponents: \( a^m \times a^n = a^{m+n} \).


• Quotient Rule: When dividing like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).


• Power of a Power Rule: When raising an exponent to another power, multiply the exponents: \((a^m)^n = a^{m \times n} \).


• Zero Exponent Rule: Any base raised to the power of zero equals one: \( a^0 = 1 \), provided that \( a eq 0 \).

Understanding and applying these rules is critical when dealing with complex expressions. They allow us to simplify terms effectively, especially when dealing with roots and radicals, as we'll see in the next sections.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This makes complex mathematical operations more manageable. When simplifying expressions with roots, you can often break them down using exponent rules.Consider the expression \( \frac{\sqrt[5]{64 x^{6}}}{\sqrt[5]{2 x}} \). By using the rules of exponents, we can rewrite it as \( \frac{(64 x^{6})^{1/5}}{(2 x)^{1/5}} \).

• First, express each term under the radical as a power with a fractional exponent. The 5th root becomes the exponent \( \frac{1}{5} \).


• Next, apply the quotient rule for exponents: \( \frac{a^m}{a^n} = a^{m-n} \). Perform the subtraction for both the numerical base and variable separately.


• Finally, simplify the remaining expression to achieve the simplest form \( 2^1 x^1 \), which is \( 2x \).

Breaking down expressions using these steps helps to simplify them effectively, leading to a much easier and more intuitive calculation process.

Roots and Radicals

Roots and radicals are fundamental in algebra, allowing us to represent the root of a number or expression. A radical is denoted by the symbol \( \sqrt{} \), and its index denotes which root is being taken.

• Square Roots: The most common radical, symbolized by \( \sqrt{x} \), usually has an implied index of 2, meaning the square root.


• Cube Roots: Represented as \( \sqrt[3]{x} \), means the third root or cube root of \( x \).


• nth Roots: More generally, the \( n \)-th root is expressed as \( \sqrt[n]{x} \), where \( n \) can be any integer.

When dealing with radicals and simplifying root expressions, converting them into expressions involving fractional exponents can make calculations much easier. For instance, the 5th root of \( x \) can be expressed as \( x^{1/5} \), which allows us to use the rules of exponents for further simplification.This conversion is particularly useful in algebra as it opens up the range of algebraic operations that can be performed, from simplification to complex expression evaluation. It also makes applying the previously mentioned exponent rules more straightforward, culminating in clearer and more elegant solutions.