Problem 81 - Algebra and Trigonometry

Question

\(69-82\) . Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. $$ \sqrt[3]{\mathcal{W}_{y}} $$

Step-by-Step Solution

Answer

\( \mathcal{W}_{y}^{1/3} \)

Step 1: Recognize the Expression

The given expression is a cube root: \( \sqrt[3]{\mathcal{W}_{y}} \). A cube root can be rewritten with a rational exponent.

Step 2: Convert Cube Root to Rational Exponent

The cube root \( \sqrt[3]{\mathcal{W}_{y}} \) can be rewritten using a rational exponent as \( \mathcal{W}_{y}^{1/3} \). This is due to the property that the \( n \)-th root of a number is equivalent to raising the number to the power of \( 1/n \).

Step 3: Simplify the Expression

In this specific exercise, rewriting the cube root as a rational exponent is all that is required. There are no further simplifications necessary.

Key Concepts

Understanding Cube Roots

A cube root is a special type of root in mathematics. It is the opposite of cubing a number. Finding the cube root of a number means determining which number multiplied by itself three times gives the original number. For example, the cube root of 8 is 2, because 2 x 2 x 2 equals 8.

In the context of the given expression, the cube root is represented as \( \sqrt[3]{\mathcal{W}_y} \). This notation tells us we are looking for a value which, when cubed, yields \( \mathcal{W}_y \).

• Cube roots help solve equations of the form \( x^3 = a \).
• They are useful in simplifying expressions and equations.

Introduction to Exponentiation

Exponentiation is a fundamental operation in mathematics that involves raising a number, known as the base, to the power of an exponent. When we raise a number to a power, we multiply it by itself as many times as indicated by the exponent.

In our exercise, converting the cube root into a rational exponent involves using the property that any \( n \)-th root of a number can be written as that number raised to the power of \( 1/n \). Thus, \( \sqrt[3]{\mathcal{W}_y} \) is expressed as \( \mathcal{W}_y^{1/3} \).

Here’s why exponentiation is important:

• It makes complex calculations simpler and more intuitive.
• Eases the manipulation and simplification of expressions.
• Forms the basis for understanding higher-level mathematical concepts.

Rational Expressions in Mathematics

Rational expressions are fractions in which the numerator and the denominator are polynomials. They are similar to regular fractions but involve variables.

In our exercise, the conversion to \( \mathcal{W}_y^{1/3} \) from \( \sqrt[3]{\mathcal{W}_y} \) yields an expression that has a rational exponent. Rational exponents share a close relationship with rational expressions in the way they simplify algebraic manipulation.

Learning about rational expressions include:

• Understanding how to simplify and combine these expressions.
• Learning how to perform operations like addition, subtraction, multiplication, and division.
• Identifying and solving equations that involve rational expressions.

This foundational understanding is key to exploring more advanced topics in algebra and calculus. Rational expressions, paired with rational exponents, offer significant flexibility and power in the manipulation of mathematical expressions.