Problem 58 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\frac{\sqrt[3]{12 x y}}{\sqrt[3]{3 x^{2} y^{5}}}\)

Step-by-Step Solution

Answer

\( \frac{\sqrt[3]{4}}{y \cdot \sqrt[3]{xy}} \)

Step 1: Simplify the Expression

Start by expressing the radicals using fractional exponents. The expression becomes: \( \frac{(12xy)^{1/3}}{(3x^2y^5)^{1/3}} \). Since both the numerator and the denominator have the same root, we can combine them under a single root: \( \sqrt[3]{\frac{12xy}{3x^2y^5}} \).

Step 2: Simplify Inside the Radicand

Simplify the fraction inside the radical. Divide the coefficients: \( \frac{12}{3} = 4 \). Then for the variables, use the property \( \frac{a^m}{a^n} = a^{m-n} \). So for x: \( \frac{x}{x^2} = x^{-1} \), and for y: \( \frac{y}{y^5} = y^{-4} \). Now we have: \( \sqrt[3]{4x^{-1}y^{-4}} \).

Step 3: Simplify Negative Exponents

Rewrite the negative exponents as reciprocals: \( x^{-1} = \frac{1}{x} \) and \( y^{-4} = \frac{1}{y^4} \). Substitute back into the expression: \( \sqrt[3]{\frac{4}{xy^4}} \).

Step 4: Write as a Single Fraction

Convert the expression into a single fraction under cube root: \( \sqrt[3]{4} \cdot \sqrt[3]{\frac{1}{xy^4}} = \frac{\sqrt[3]{4}}{\sqrt[3]{x} \cdot \sqrt[3]{y^4}} \).

Step 5: Further Simplify the Radicals

Since \( y^4 = y^3 \cdot y \), its cube root becomes: \( \sqrt[3]{y^4} = y \cdot \sqrt[3]{y} \). Thus the expression is: \( \frac{\sqrt[3]{4}}{\sqrt[3]{x} \cdot y \cdot \sqrt[3]{y}} \).

Step 6: Final Simplification

Combine the cube roots: \( \frac{\sqrt[3]{4}}{y} \cdot \frac{1}{\sqrt[3]{xy}} \). The final simplest radical form is: \( \frac{\sqrt[3]{4}}{y \cdot \sqrt[3]{xy}} \).

Key Concepts

Fractional Exponents

Fractional exponents, also known as rational exponents, are a way to express roots in a different form. For example, if you have a cube root of a number, it can be written with a fractional exponent of \( \frac{1}{3} \). This means the same as the cube root: \( a^{1/3} = \sqrt[3]{a} \).When simplifying expressions with fractional exponents, you can often combine or simplify the terms more easily. Adding, subtracting, or combining roots can be made simpler by converting the root expressions into these fractional exponent forms. For instance, \( (12xy)^{1/3} \) represents the cube root of \(12xy\), and helps in manipulating algebraic expressions.

Converts roots into easier-to-manage forms

Helps in algebraic manipulation and simplification

Using fractional exponents, you can work through mathematical expressions step-by-step, transforming complex roots into simpler, more calculable factors during problem-solving.

Cube Roots

A cube root is the number that, when multiplied by itself three times, returns the original number. Calculating cube roots can help in solving equations where the power of a variable or a constant is three, such as \( \sqrt[3]{a} \).In our exercise, simplifying the cube root involves both the coefficients and the variables. For example, to find \( \sqrt[3]{4x^{-1}y^{-4}} \), each element inside the root sign is individually processed under cube roots.

• Cube roots are inverse operations of cubing numbers
• Used to simplify expressions under radical signs

Cube roots are frequently encountered in polynomial equations and various algebraic problems, playing a crucial role in simplifying radical forms.

Algebraic Fractions

Algebraic fractions are fractions wherein the numerator, denominator, or both contain algebraic expressions. To simplify them, we often need to reduce the fractions just like regular fractions, by finding common factors or applying exponent rules.In this context, our task is to simplify \( \frac{12xy}{3x^2y^5} \). We accomplish that by dividing both numerical coefficients and the variables with exponent rules, ultimately reducing them to \( \frac{4}{xy^4} \).

Simplification involves basic arithmetic and exponent rules

The goal is often to express in simplest terms for easier understanding

Simplification of algebraic fractions helps make complex expressions easier to handle, especially when they are part of larger equations or functions.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive of that exponent. For example, \( a^{-n} = \frac{1}{a^n} \).Handling negative exponents in our given problem involves rewriting expressions such as \( x^{-1} \) and \( y^{-4} \) into their reciprocal forms. Thus, \( x^{-1} = \frac{1}{x} \) and \( y^{-4} = \frac{1}{y^4} \).

• Transform expressions to positive exponents to simplify operations
• Use reciprocal properties for straightforward calculations

Rewriting negative exponents into positive expressions allows for the simplification of radicands, enabling problem solvers to manage and simplify complex equations efficiently.