Problem 74 - Algebra A Combined Function

Question

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt[4]{160 x^{10} y^{5}}}{\sqrt[4]{2 x^{2} y^{2}}}\)

Step-by-Step Solution

Answer

The simplified form is \( 2x^{2} y^{3/4} \sqrt[4]{5} \).

Step 1: Apply the Quotient Rule for Roots

The quotient rule for roots states that \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \). Thus, \( \frac{\sqrt[4]{160 x^{10} y^{5}}}{\sqrt[4]{2 x^{2} y^{2}}} = \sqrt[4]{\frac{160 x^{10} y^{5}}{2 x^{2} y^{2}}} \).

Step 2: Simplify the Fraction Under the Root

Divide the terms inside the root: \( \frac{160 x^{10} y^{5}}{2 x^{2} y^{2}} = \frac{160}{2} \cdot \frac{x^{10}}{x^{2}} \cdot \frac{y^{5}}{y^{2}} = 80 x^{8} y^{3} \).

Step 3: Evaluate the Fourth Root

Evaluate the fourth root of each term individually: \( \sqrt[4]{80 x^{8} y^{3}} = \sqrt[4]{80} \cdot \sqrt[4]{x^{8}} \cdot \sqrt[4]{y^{3}} \).

Step 4: Simplify the Individual Roots

Note that \( \sqrt[4]{x^{8}} = x^{2} \) because \( x^{8} = (x^{2})^{4} \). The root of \( y^{3} \) remains \( (y^{3})^{1/4} \). For their coefficients, break down 80 into its factors: \( 80 = 16 \times 5 \), so \( \sqrt[4]{80} = \sqrt[4]{16 \cdot 5} = \sqrt[4]{16} \cdot \sqrt[4]{5} = 2 \cdot \sqrt[4]{5} \).

Step 5: Combine Simplified Components

Put all the simplified components together: \( 2 \cdot x^{2} \cdot \sqrt[4]{5} \cdot y^{3/4} \). Thus, the simplified expression is \( 2x^{2} y^{3/4} \sqrt[4]{5} \).

Key Concepts

Fourth Root Simplification

Simplifying expressions with roots can often seem tricky, but it becomes clearer when we break it down into manageable parts. The fourth root of a number or expression indicates that the expression is raised to the power of one-fourth. When simplifying

1. identify any factors or terms that are perfect fourth powers
2. snoop for opportunities to divide the original numbers or expressions in a manner that simplifies the root.

In the context of the given exercise, note that simplifying the fourth root of a fraction is made easier by using the quotient rule. It allows us to rework a division inside a root into a root of a division, simplifying both the numerator and the denominator.

For instance: - The number 80 can be broken down into factors like 16 (a perfect fourth power) and 5. - Similarly, expressions under the root, like 80, can be separated into smaller components to simplify the calculation. Understanding how to identify these factors makes the simplification process quicker and clearer.

Variables in Algebra

Handling variables in algebra requires understanding how they behave under various operations. A variable in an expression represents an unknown value, and in operations like division or root simplification, it's crucial to merge similar terms.

In expressions like \(\frac{x^{10}}{x^{2}}\), the exponents are handled using exponent rules. Exponent division can be simplified by subtracting exponents:

• \(x^{10}/x^{2} = x^{10-2} = x^{8}\)

Apply similar rules to the coefficient of the variables. Here is a basic approach:

For terms with the same base:

• Variables such as \( y^{5} / y^{2} = y^{5-2} = y^{3}\)

When dealing with roots and other operations, approach the problem with clarity on basic algebra rules to simplify quicker and reduce mistakes. Recognize when to apply each rule and make each step count.

Expression Simplification

Expression simplification in algebra embraces the process of making an expression as concise as possible without losing its original value. The goal is to make it more understandable and usable. In the exercise provided, several steps were involved in simplifying the expression to eventually arrive at a clearer form:

1. **Breaking down coefficients**: - Simplify numerical components to make it handy while handling roots or powers.2. **Factoring exponents**: - Dividing variables by subtracting their exponents often thins down parts of the expression.3. **Using root simplification**: - Especially when dealing with higher-order roots, know when a term's exponent is easily extractable, and when you’re left with a leftover fractional power:

• For example, \(x^{8}\) becomes \((x^2)^4\) under the fourth root \(\rightarrow x^2\)
• Parts that can't be further simplified are kept as is, such as \(\sqrt[4]{5}\).

In the end, combining the simplified parts—coefficients, variables, and roots—results in an expression that retains all original values but is much neater. Thus, reaching a form like \(2x^{2} y^{3/4} \sqrt[4]{5}\), where every factor is crystal clear in its simplicity.