Problem 56 - Precalculus Mathematics for Calculus

Question

Simplify the expression. Assume that the letters denote any real numbers. $$\sqrt[3]{x^{3} y^{6}}$$

Step-by-Step Solution

Answer

The simplified expression is \( xy^2 \).

Step 1: Recognize the Cube Root

The expression given is a cube root. This means that we need to find the number that, when raised to the power of 3, would give us the original expression inside the root.

Step 2: Apply the Cube Root to Each Factor

Break down the expression \ \( \sqrt[3]{x^{3} y^{6}} \ \) into separate cube roots for each factor: \ \( \sqrt[3]{x^{3}} \) and \ \( \sqrt[3]{y^{6}} \). The property \ \( \sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b} \) allows us to simplify complex roots step-by-step.

Step 3: Simplify the Cube Root of Powers

Use the property that \ \( \sqrt[3]{a^n} = a^{n/3} \): \begin{align*} \sqrt[3]{x^3} & = x^{3/3} = x, \ \sqrt[3]{y^6} & = y^{6/3} = y^2. \end{align*}

Step 4: Combine the Results

Now that we have simplified each cube root, combine them: The expression \ \( \sqrt[3]{x^{3} y^{6}} \) simplifies to \ \( xy^2 \).

Key Concepts

Exponents

Exponents are a fundamental concept in algebra that represent repeated multiplication of a number by itself. For instance, when you see an exponent like 3 in the expression \( x^3 \), it indicates that \( x \) is multiplied by itself three times: \( x \times x \times x \). This makes calculations much more manageable and notation more concise.Understanding how to manipulate exponents can simplify many algebraic expressions. Some important properties include:

• Product of Powers: If you multiply terms with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Power: When raising an exponent to another power, you multiply the exponents: \( (a^m)^n = a^{m \times n} \).
• Power of a Product: Distribute the exponent to each part of the product: \( (ab)^n = a^n b^n \).

These rules are crucial when dealing with expressions involving roots, such as cube roots, since they help us rewrite the terms in a form that's easier to simplify.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. These are expressions that require finding a number which, when raised to a certain power, yields the expression inside the radical sign. For example, the cube root of \( x^3 \) is the number that, when cubed, equals \( x^3 \). When simplifying radical expressions, there are vital steps and properties to remember:

• Breaking Down Roots: The property \( \sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} \) helps in breaking down complex products into simpler terms to handle each part individually.
• Simplification of Powers: Use the property \( \sqrt[n]{a^m} = a^{m/n} \) to turn roots into fractional exponents, making it easier to apply exponent rules.

By mastering these principles, working with radicals becomes a much more straightforward endeavor.

Simplification of Algebraic Expressions

Simplifying algebraic expressions is about reducing them to their simplest form while retaining the same value. For the cube root expression \( \sqrt[3]{x^{3} y^{6}} \), simplification involves breaking it down and utilizing exponent rules efficiently.Here’s how to simplify:

• Recognize the parts of the expression, like \( x^3 \) and \( y^6 \), and separately simplify them using cube roots and exponent rules.
• Convert the radical expression into one with fractional exponents: \( x^{3/3} \) and \( y^{6/3} \).
• Simplify the fractions \( 3/3 \) and \( 6/3 \), resulting in \( x^1 \) and \( y^2 \).
• Combine simplified parts back into a single expression: \( xy^2 \).

These steps turn complex equations into manageable forms and are essential for efficient problem-solving in algebra.