Problem 74 - Elementary and Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt[3]{9 x^{5} y^{4}}}{\sqrt[3]{3 x^{5} y^{5}}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{\sqrt[3]{3 y^{2}}}{y} \).

Step 1: Rewrite the Expression

Start by rewriting the given expression in a fractional form to make it clearer. The given expression is \( \frac{\sqrt[3]{9 x^{5} y^{4}}}{\sqrt[3]{3 x^{5} y^{5}}} \). You can express this as a single cube root:\[\sqrt[3]{\frac{9 x^{5} y^{4}}{3 x^{5} y^{5}}}\]Now, simplify the expression inside the cube root.

Step 2: Simplify the Inside of the Cube Root

Simplify the expression inside the cube root by dividing the numerator's terms by the denominator's terms:- For the constants, \( \frac{9}{3} = 3 \).- For \( x^{5} \) terms, \( \frac{x^{5}}{x^{5}} = 1 \).- For \( y^4 \) and \( y^5 \), \( \frac{y^4}{y^5} = \frac{1}{y} \).So the expression becomes:\[\sqrt[3]{3 \cdot \frac{1}{y}} = \sqrt[3]{\frac{3}{y}}\]

Step 3: Rationalize the Denominator

To rationalize the denominator, we need to get rid of the cube root in the denominator by multiplying by a form of 1 that will eliminate the cube root. Multiply both the numerator and the denominator inside the cube by \( y^{2} \):\[\sqrt[3]{\frac{3 \, y^{2}}{y \, y^{2}}} = \sqrt[3]{\frac{3 \, y^{2}}{y^{3}}}\]The denominator becomes \( y^{3} \) which is a perfect cube and simplifies directly.

Step 4: Simplify the Cube Root

The cube root of \( y^3 \) is \( y \). So the expression simplifies to:\[\frac{\sqrt[3]{3 \cdot y^2}}{\sqrt[3]{y^3}} = \frac{\sqrt[3]{3 \, y^{2}}}{y}\]This is the expression with the denominator rationalized.

Key Concepts

Cube Roots

When working with cube roots, you're essentially finding a number that, when multiplied by itself three times, gives you the original number. Cube roots are denoted using the radical symbol with a small "3" (\( \sqrt[3]{...} \)). For example, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \).

Cube roots can be very useful in simplifying expressions, especially when dealing with division. In the context of our exercise, we are dealing with cube roots of fractional base numbers. Simplifying within a cube root might involve dividing the interior terms, just as you would outside of a root.

• For example, if you have a cube root of \( \frac{27}{8} \), you can separate this into \( \frac{\sqrt[3]{27}}{\sqrt[3]{8}} = \frac{3}{2} \).

Understanding cube roots is crucial when dealing with terms under the radical, making it easier to rationalize and simplify the expressions you come across.

Exponent Laws

Exponent laws are the rules that govern the operations on numbers with exponents. These laws help when simplifying expressions that include variables raised to powers. In the context of rationalizing denominators or simplifying cube roots, knowing these laws can be a game changer.

• Product of powers rule: When you multiply like bases, add the exponents: \( a^{m} \cdot a^{n} = a^{m + n} \).
• Quotient of powers rule: For dividing like bases, subtract the exponents: \( \frac{a^{m}}{a^{n}} = a^{m-n} \).
• Power of a power rule: When raising an exponential to another power, multiply the exponents: \( (a^{m})^{n} = a^{m \cdot n} \).

In this exercise, the quotient of powers rule is particularly useful. For example, \( \frac{x^{5}}{x^{5}} = x^{5-5} = x^{0} = 1 \). Such simplifications help us break down the complexity in expressions and tackle more manageable components while rationalizing denominators.

Simplifying Expressions

Simplifying expressions makes them easier to work with, especially when they become part of more complex operations, like rationalizing denominators. It involves several key steps: breaking down the expression, applying arithmetic and algebraic rules, and reducing it to its simplest form.

In this problem, we start by transforming the expression under the cube root to a simpler form. By dividing each term in the numerator by the corresponding term in the denominator, we reduced the complexity significantly.

• First, address the constants by division. For instance, \( \frac{9}{3} = 3 \).
• Next, use exponent laws for variables, such as \( \frac{y^{4}}{y^{5}} = y^{4-5} = \frac{1}{y} \).

After obtaining a simpler expression under the radical, it's then easier to multiply the numerator and denominator within the cube root by terms necessary to rationalize the denominator. Doing so ultimately leads to a clear, simplified outcome. It's all about gradually breaking down each part to reveal a neat and tidy result.