Problem 72 - Algebra A Combined Function

Question

Assume that all variables represent positive real numbers. $$ \sqrt[4]{x^{8} y^{12}} $$

Step-by-Step Solution

Answer

The simplified expression is \(x^2 y^3\).

Step 1: Simplify the Fourth Root

To simplify the expression \( \sqrt[4]{x^8 y^{12}} \), we begin by recognizing the properties of exponents and roots. We can express the fourth root as raising to the power of \( \frac{1}{4} \). Thus, \( \sqrt[4]{x^8 y^{12}} = (x^8 y^{12})^{\frac{1}{4}} \).

Step 2: Apply Power Rule to Exponents

Apply the power rule \( (a^m)^n = a^{m \cdot n} \). We rewrite the expression \((x^8 y^{12})^{\frac{1}{4}}\) as \(x^{8 \cdot \frac{1}{4}} y^{12 \cdot \frac{1}{4}}\).

Step 3: Multiply Exponents

Calculate the products of the exponents: \(8 \cdot \frac{1}{4} = 2\) and \(12 \cdot \frac{1}{4} = 3\). Thus, the expression simplifies to \(x^2 y^3\).

Key Concepts

Fourth Root

The concept of the fourth root can sometimes seem daunting but is actually quite straightforward. A fourth root of a number is the value that, when multiplied by itself four times, equals the original number. Mathematically, the fourth root of a number can be expressed as raising the number to the power of \( \frac{1}{4} \).
For example, if we have a number \( n \), the fourth root is \( n^{\frac{1}{4}} \).

• In our given exercise, we simplify \( \sqrt[4]{x^8 y^{12}} \), which is equivalent to \( (x^8 y^{12})^{\frac{1}{4}} \).
• This step involves recognizing that the fourth root operation is essentially applying the exponent \( \frac{1}{4} \) to the entire expression inside the root.
• It turns a root problem into an exponent problem, making it easier to handle when combined with other exponent rules.

Properties of Exponents

Understanding the properties of exponents is crucial when simplifying mathematical expressions involving powers. These properties allow us to efficiently manipulate and simplify expressions. Here are the basic ones that apply to the exercise:

• Product of Powers Rule: When multiplying two exponential expressions with the same base, you add the exponents, i.e., \( a^m \times a^n = a^{m+n} \).
• Power of a Power Rule: This rule states that when you have an exponent raised to another exponent, you multiply the exponents, i.e., \( (a^m)^n = a^{m\cdot n} \).
• Power of a Product Rule: To apply the power to a product of bases, you distribute the power to each factor inside the parenthesis, i.e., \( (ab)^m = a^m b^m \).

In our exercise, we use these properties to break down and simplify the expression \( (x^8 y^{12})^{\frac{1}{4}} \), by applying the exponent \( \frac{1}{4} \) to both \( x^8 \) and \( y^{12} \). This simplifies our problem into a product of simpler terms.

Power Rule

The power rule is an essential tool in simplifying expressions involving exponents. It helps transform complex exponent expressions into simpler forms. Specifically, it states: \((a^m)^n = a^{m \cdot n}\).
In the context of the current exercise, this rule is used to further simplify the problem:

• We start with the expression \( (x^8 y^{12})^{\frac{1}{4}} \).
• Applying the power rule to each component separately, we make use of \((x^8)^{\frac{1}{4}} = x^{8 \cdot \frac{1}{4}}\) and \((y^{12})^{\frac{1}{4}} = y^{12 \cdot \frac{1}{4}}\).
• This results in \( x^2 y^3 \) after multiplying the exponents, where \( 8 \cdot \frac{1}{4} = 2 \) and \( 12 \cdot \frac{1}{4} = 3 \).

The power rule simplifies the process of breaking down and managing the exponentiation, allowing us to turn a compound exponent into a product of simpler powers.