Problem 76 - Algebra A Combined Function

Question

Use the quotient rule to divide. Then simplify if possible. Assume that all variables represent positive real numbers. \(\frac{\sqrt[5]{192 x^{6} y^{12}}}{\sqrt[5]{2 x^{-1} y^{-3}}}\)

Step-by-Step Solution

Answer

2xy^3 \cdot \sqrt[5]{3x^2}

Step 1: Apply the Quotient Rule for Radicals

According to the quotient rule for radicals, \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \). Here, \( n = 5 \), so we write: \[ \sqrt[5]{\frac{192 x^{6} y^{12}}{2 x^{-1} y^{-3}}} \]

Step 2: Simplify the Expression Inside the Radical

Start by simplifying the terms inside the radicals. Divide the constants, and apply the properties of exponents to simplify the expression:\[ \frac{192}{2} = 96 \]\[ \frac{x^{6}}{x^{-1}} = x^{6 - (-1)} = x^{7} \]\[ \frac{y^{12}}{y^{-3}} = y^{12 - (-3)} = y^{15} \] Hence, the simplified expression becomes:\[ \sqrt[5]{96 x^{7} y^{15}} \]

Step 3: Simplify the Radicals When Possible

Now, attempt to simplify inside the radical. If possible, express the radicals in the form where factors are perfect powers of 5:- For 96, the factors are: \( 2^5 \cdot 3 \). Thus, \( 96 = (2^5) \cdot 3 \)- \( x^7 = x^{5}\cdot x^{2} \)- \( y^{15} = (y^{5})^3 \)This simplifies to:\[ \sqrt[5]{2^5 \cdot 3 \cdot x^5 \cdot x^2 \cdot (y^5)^3 } \]

Step 4: Simplifying Further Using the Properties of Radicals

Extract the perfect fifth powers:- \( \sqrt[5]{2^5} = 2 \)- \( \sqrt[5]{x^5} = x \)- \( \sqrt[5]{(y^5)^3} = y^3 \)This gives us:\[ 2xy^3 \cdot \sqrt[5]{3x^2} \]Thus, the entire expression becomes:\[ 2xy^3 \cdot \sqrt[5]{3x^2} \]

Step 5: Conclusion

The simplified expression for the given problem using the quotient rule is \( 2xy^3 \cdot \sqrt[5]{3x^2} \).

Key Concepts

Understanding Radicals

In mathematics, radicals are symbols that denote the root of a number. They are often represented by a radical sign, \( \sqrt{} \), followed by a number inside, called the radicand. In the exercise, we deal specifically with the fifth root, which is a type of radical.

• The fifth root of a number \( a \) is a number that, when multiplied by itself five times, gives the number \( a \).
• In expression terms, the fifth root of \( a \) is written as \( \sqrt[5]{a} \).

Radicals, especially higher-order ones like the fifth root, are useful in calculations involving complex expressions. They help simplify and solve these expressions into more manageable forms.

The Process of Simplification

Simplification is about making mathematical expressions less complex. The aim is to transform them into a more basic, easily understandable form. When dealing with radicals, simplification involves:

• Combining similar terms within the radicals.
• Breaking down numbers into their prime factors to find any perfect powers, if possible.

In the exercise, we simplified by: dividing constants, using properties of exponents to simplify variables, and identifying perfect fifth powers. This step is crucial as it helps reveal simpler forms of expressions for easy calculations in further steps.

Properties of Exponents

Exponents denote the number of times a number (the base) is multiplied by itself. When working with radicals and quotient rules, properties of exponents are essential for simplification:

• Product of Powers Rule: \( x^a \cdot x^b = x^{a+b} \)
• Quotient of Powers Rule: \( \frac{x^a}{x^b} = x^{a-b} \)
• Power of a Power Rule: \( (x^a)^b = x^{a \cdot b} \)

In the provided exercise, we applied these rules to simplify expressions like \(\frac{x^6}{x^{-1}}\), converting it to more manageable powers such as \(x^7\). Understanding and applying these rules helps untangle complicated exponents within radicals.

Exploring the Fifth Root

The fifth root specifically refers to taking the fifth power of a number or expression and reversing it. It is especially relevant when simplifying expressions in radicals, as demonstrated in the exercise.

• The fifth root is not just about simplifying expressions but also finding factors that are raised to the fifth power.
• If an expression is a perfect fifth power, it can be simplified directly, making further calculations straightforward.

For instance, in our exercise, identifying parts of the expression such as \(2^5\), \(x^5\), and \(y^5\) allowed us to simplify directly to individual terms (like 2, \(x\), and \(y^3\)), simplifying the overall expression into \(2xy^3 \cdot \sqrt[5]{3x^2}\). These transformations make dealing with larger, more complex expressions more approachable.