Problem 59 - 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

The simplest radical form is \(\frac{\sqrt[3]{4}}{x^{1/3} y^{4/3}}\).

Step 1: Write the Expression in Radical Form

The given expression is in the form of a fraction with cube roots. It can be written as: \[\frac{\sqrt[3]{12xy}}{\sqrt[3]{3x^2y^5}} = \sqrt[3]{\frac{12xy}{3x^2y^5}}\]

Step 2: Simplify the Fraction inside the Cube Root

Simplify the expression inside the cube root by dividing each part of the fraction:\[\frac{12xy}{3x^2y^5} = \frac{12}{3} \times \frac{x}{x^2} \times \frac{y}{y^5}\]

Step 3: Perform the Division for Each Part

Divide each component:- \(\frac{12}{3} = 4\)- \(\frac{x}{x^2} = \frac{1}{x}\)- \(\frac{y}{y^5} = \frac{1}{y^4}\)So, the fraction simplifies to: \[4 \times \frac{1}{x} \times \frac{1}{y^4} = \frac{4}{xy^4}\]

Step 4: Write the Simplified Expression inside the Cube Root

Now, the expression inside the cube root is:\[\sqrt[3]{\frac{4}{xy^4}}\]

Step 5: Simplify the Cube Root

We can simplify the cube root as follows:- \(\sqrt[3]{4} = \sqrt[3]{4}\)- \(\sqrt[3]{\frac{1}{x}} = \frac{1}{\sqrt[3]{x}}\)- \(\sqrt[3]{\frac{1}{y^4}} = \frac{1}{y \cdot \sqrt[3]{y}}\)Thus, the expression becomes:\[\frac{\sqrt[3]{4}}{x^{1/3} y^{4/3}}\]

Step 6: Write Final Answer in Simplified Radical Form

The simplest radical form of the given expression is: \[\frac{\sqrt[3]{4}}{x^{1/3} y^{4/3}}\]

Key Concepts

Radical Form

The concept of a radical form involves expressing numbers using roots. In this context, it involves cube roots, because we are dealing with the expression \( \sqrt[3]{12xy} \). The radical sign \( \sqrt{} \) with an index of 3 indicates a cube root.
In radical form, we simplify expressions by ensuring the values inside the radical sign are reduced to their basic components. So for a cube root, we aim to express the number inside using the smallest factors possible.
In the initial exercise, the expression in radical form was an essential first step because this allows easier manipulation and simplification using properties of exponents and algebraic identities.

Cube Roots

Cube roots focus on finding a number, when multiplied by itself three times, equals the given number. For instance, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \).
In mathematical notation, a cube root is represented as \( \sqrt[3]{a} \). To simplify cube roots, especially when they involve variables and fractions like \( \sqrt[3]{\frac{12xy}{3x^2y^5}} \), it's vital to look at the composition of the numbers and variables inside.

• Reduce the coefficients, such as simplifying \( \frac{12}{3} \) to 4.
• Apply the laws of exponents: for example, \( \frac{x}{x^2} = x^{1-2} = x^{-1} \).
• Simplify systematically for each part of the expression, considering each variable and exponent separately inside the cube root.

Simplifying Fractions

When simplifying fractions, our goal is to reduce the fraction to its smallest possible form. This involves canceling out common factors in the numerator and the denominator. For the exercise example \( \frac{12xy}{3x^2y^5} \), this is executed as follows:

• Divide numerical coefficients, so \( \frac{12}{3} \) becomes 4.
• For variable terms, use the property \( \frac{x^a}{x^b} = x^{a-b} \). For instance, \( \frac{x}{x^2} = x^{-1} \).
• With multiple terms like \( \frac{y}{y^5} \), apply exponents to get \( y^{-4} \).

By breaking it down this way, we simplify the expression to \( \frac{4}{xy^4} \), making it easier to work with inside the radical.

Variable Exponents

Variable exponents are numbers or variables raised to the power of another number, such as \( x^2 \). Understanding how to manipulate them is vital for simplifying expressions involving radicals and fractions.
In this problem, variable exponents involved operations like \( x^{1-2} \) and \( y^{1-5} \). Here, we subtract exponents of like bases:

• \( x^a / x^b = x^{a-b} \) - this rule helps in simplifying power terms.
• Negative exponents indicate division. For example, \( x^{-1} = \frac{1}{x} \) and \( y^{-4} = \frac{1}{y^4} \).

Applying these rules helps us rewrite expressions inside radicals in a form we can easily interpret and further simplify. Variable exponents can make expressions look complex, but these simplification techniques pave the way to a clearer solution.