Problem 74 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[5]{\frac{3 x^{11} y^{3}}{9 x^{2}}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x^{9/5} y^{3/5} 3^{4/5}}{3} \).

Step 1: Simplify the Fraction

Start by simplifying the expression inside the radical. The expression is \( \frac{3x^{11}y^3}{9x^2} \). First, simplify the coefficients: \( \frac{3}{9} = \frac{1}{3} \). Next, simplify the powers of \( x \): \( \frac{x^{11}}{x^{2}} = x^{11-2} = x^{9} \). So the expression becomes \( \frac{x^9 y^3}{3} \).

Step 2: Apply the Fifth Root to the Numerator

Next, apply the fifth root to each part of the numerator separately: \( \sqrt[5]{x^9 y^3} \). By the property of exponents \((a^m)^n = a^{m \cdot n}\), we have \( \sqrt[5]{x^9} = x^{\frac{9}{5}} \) and \( \sqrt[5]{y^3} = y^{\frac{3}{5}} \).

Step 3: Apply the Fifth Root to the Denominator

Apply the fifth root to the denominator separately: \( \sqrt[5]{3} \). Since \( 3 \) doesn't simplify easily, we leave it as it is.

Step 4: Combine under Fifth Root

Combine the fifth roots for the fraction: \( \frac{\sqrt[5]{x^9 y^3}}{\sqrt[5]{3}} = \sqrt[5]{\frac{x^9 y^3}{3}} \). Now, separate the expression into: \( \frac{x^{9/5} y^{3/5}}{3^{1/5}} \).

Step 5: Rationalize the Denominator

To rationalize the denominator, multiply the numerator and the denominator by \( 3^{4/5} \) to make the power of \( 3 \) in the denominator equal to 1: \( \frac{x^{9/5} y^{3/5} \cdot 3^{4/5}}{3^{1/5} \cdot 3^{4/5}} = \frac{x^{9/5} y^{3/5} 3^{4/5}}{3} \).

Step 6: Final Simplified Expression

The expression simplifies to \( x^{9/5} y^{3/5} 3^{4/5} / 3 \) with a rationalized denominator.

Key Concepts

Simplifying Radicals

Radicals are expressions that involve roots, like square roots or fifth roots. When you simplify radicals, your goal is to make them as simple and clear as possible.
In our exercise, you start by simplifying the entire expression inside the fifth root. This involves simplifying both the numerator and the denominator separately.

• First, simplify any coefficients. Divide numbers if possible. In our case, \( \frac{3}{9} = \frac{1}{3} \).
• Next, simplify the variables by using exponent rules. Here, \((x^{11})/(x^2) = x^{11-2} = x^9\)

After simplifying, you end up with an easier expression to work with before applying the radical. This makes following operations more straightforward.

Exponent Rules

Dealing with radicals often involves handling exponents since radicals can be expressed as fractional exponents. Knowing exponent rules helps to simplify such expressions.

Take for example the term \( x^9 y^3\) inside the radical in our problem. When you apply a fifth root, you are essentially changing this to fractional exponents:

• The fifth root of \( x^9 \) is \( x^{9/5}\).
• The fifth root of \( y^3 \) is \( y^{3/5}\).

These transformations use the property \( (a^m)^n = a^{m \cdot n} \). Converting radicals into exponents can simplify the calculation by allowing you to apply multiplication and division of exponents rules easily.

Exponent rules are powerful in algebra because they help in breaking down expressions and bridging steps between problems and solutions.

Rationalizing the Denominator

Rationalizing the denominator is an important process, especially when dealing with radicals. It involves changing the denominator of a fraction to a rational number.

In our case, the denominator contains \( 3^{1/5} \). To get rid of the radical, you multiply both the numerator and denominator by \( 3^{4/5} \). This multiplication effectively removes the radical from the denominator, as multiplying exponents with the same base involves adding the exponents:

• The denominator becomes \( 3^{1/5 + 4/5} = 3^1 = 3\), which is rational.
• The numerator gets multiplied by \( 3^{4/5} \), turning into \( x^{9/5} y^{3/5} \cdot 3^{4/5}\).

By rationalizing the denominator, you make the expression cleaner and easier to handle, especially when further algebraic manipulation is needed, like in mathematical proofs or additional calculations.