Problem 75 - Algebra A Combined Function

Question

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt[5]{64 x^{10} y^{3}}}{\sqrt[5]{2 x^{3} y^{-7}}}\)

Step-by-Step Solution

Answer

The simplified expression is: \(2xy^2 \sqrt[5]{x^2}\).

Step 1: Set Up the Quotient Rule for Radicals

When dividing radicals with the same index, divide the radicands and keep the same index. The expression is: \[\frac{\sqrt[5]{64 x^{10} y^{3}}}{\sqrt[5]{2 x^{3} y^{-7}}} = \sqrt[5]{\frac{64 x^{10} y^{3}}{2 x^{3} y^{-7}}}\]

Step 2: Simplify the Radicand

Divide the coefficients, and apply the division of exponents rule for each variable inside the radical: \[\frac{64}{2} = 32, \quad \frac{x^{10}}{x^{3}} = x^{7}, \quad \frac{y^{3}}{y^{-7}} = y^{3 - (-7)} = y^{10}\] Thus, the simplified radicand becomes:\[32 x^7 y^{10}\]

Step 3: Rewrite as a Single Radical Expression

Insert the simplified radicand into the radical expression:\[\sqrt[5]{32 x^7 y^{10}}\]

Step 4: Simplify the Radical Expression

Identify perfect fifth powers within the radicand (if any) and simplify:- 32 is \(2^5\), so \(\sqrt[5]{32} = 2\).- \(x^7\) can be written as \(x^5 \cdot x^2\), simplifying to \(x \cdot \sqrt[5]{x^2}\).- \(y^{10}\) can be written as \(y^{10/5} = y^2\).So, the simplified expression is:\[2xy^2 \sqrt[5]{x^2}\]

Step 5: Final Answer

The final simplified expression for the given radical division is: \[2xy^2 \sqrt[5]{x^2}\]

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, or any higher-level roots. In the context of radical division, understanding how to work with radicals is key. A radical expression typically consists of a radicand (the number or expression inside the root) and an index, which indicates the degree of the root. For example, in \( \sqrt[5]{64 x^{10} y^{3}} \), the index is 5, indicating a fifth root, and the radicand is \( 64x^{10}y^3 \).
It's important to keep in mind that radicals can often be manipulated to simplify expressions. Often, operations like multiplication and division can be performed inside the radicand, due to properties of radicals. This allows for simplification before or after the actual operation.
To divide radical expressions that have the same index, as in our problem, we use the quotient rule for radicals. This rule states that the division of radicals can be performed by dividing their radicands and keeping the same index. Essentially, you create a single radical with the quotient of the radicands, streamlining the operation and simplifying the expression dramatically.

Simplifying Radicals

Simplifying radicals involves expressing the radical in its simplest form. After using the quotient rule, the focus shifts to simplifying both the numerical and the variable parts within the radicand. Let's consider the step where we simplified \( \frac{64 x^{10} y^{3}}{2 x^{3} y^{-7}} \).

• Numerical Simplification: Here, we divide the coefficients, so \( \frac{64}{2} = 32 \). This simplifies the numbers involved, making the expression simpler.
• Variable Simplification: Use the division of exponents rule, \( a^m / a^n = a^{m-n} \), to simplify the variable part:
• For \( x \), \( \frac{x^{10}}{x^{3}} = x^{7} \)
• For \( y \), \( \frac{y^{3}}{y^{-7}} = y^{3 - (-7)} = y^{10} \)

This step results in a new radicand: \( 32x^7y^{10} \).
The goal is to identify perfect powers within the radicand. For instance, \( 32 \) can be expressed as \( 2^5 \), allowing us to simplify \( \sqrt[5]{32} \) to 2. For variables, rewrite any variables in the form of powers that are multiples of the index if possible. The expression becomes simpler to handle and interpret by distilling it to its core components.

Division of Exponents

The division of exponents is crucial when simplifying expressions involving powers. The core rule to remember is \( a^m / a^n = a^{m-n} \). This rule is applied when dividing like bases with exponents. In the context of radicals and their division, it's particularly handy when simplifying the variable parts:
Applying this to our exercise:

• For \( x^{10} / x^{3} \), subtract the exponent in the denominator from the exponent in the numerator to get \( x^{10-3} = x^7 \).
• Similarly, for negative exponents like \( y^{3} / y^{-7} \), subtracting means \( y^{3 - (-7)} = y^{10} \).

This approach not only simplifies the expression, but it also makes it more manageable and easier to understand.
By understanding division of exponents, you gain the ability to handle more complex expressions and simplify them in the context of radicals effectively. This technique is fundamental in working efficiently with expressions that involve radical division.