Problem 56 - Algebra for College Students

Question

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(4 y \sqrt[3]{x}\)

Step-by-Step Solution

Answer

The expression is \(4y x^{\frac{1}{3}}\) with positive rational exponents.

Step 1: Identify the Roots

The given expression is \(4y\sqrt[3]{x}\). The term \(\sqrt[3]{x}\) denotes the cube root of \(x\).

Step 2: Apply Rational Exponent Rule

The cube root \(\sqrt[3]{x}\) can be expressed with a rational exponent as \(x^{\frac{1}{3}}\).

Step 3: Rewrite the Expression

Replace the cube root with its rational exponent form to get the expression \(4yx^{\frac{1}{3}}\).

Step 4: Complete the Solution

Combine all parts to express the original expression \(4y\sqrt[3]{x}\) using positive rational exponents, leading to the final result \(4yx^{\frac{1}{3}}\).

Key Concepts

Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because when you multiply 2 by itself three times (\(2 \times 2 \times 2\)), you get 8.
In algebra, cube roots are often expressed using the radical symbol with a small 3, like \( \sqrt[3]{x} \). This concept can be especially useful when you're looking to simplify expressions where cube roots appear.
However, in many cases, we find it convenient to represent cube roots using rational exponents for more straightforward manipulation in algebraic operations. For a given number \(a\), the cube root can be expressed as \(a^{\frac{1}{3}}\). This notation is cleaner and makes it easier to apply algebraic rules, especially when working with expressions that include variables and powers.

Rational Expressions

Rational expressions are a type of algebraic expression that involve fractions. A rational expression is simply a fraction in which both the numerator and the denominator are polynomials. For example, \(\frac{x+1}{x^2-4}\) is a rational expression, with \(x+1\) as the numerator and \(x^2-4\) as the denominator.
When dealing with rational expressions, it's important to ensure the expression is defined; this means the denominator cannot be zero. Thus, finding the domain of the expression is crucial.
Rational exponents can also appear within rational expressions. For instance, if an expression contains terms like \(x^{\frac{1}{3}}\), we treat these with the same rules we apply to polynomial terms. Operations such as addition, subtraction, multiplication, and division can be performed on rational expressions as long as common vocabulary terms like factoring and finding a common denominator are understood.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. For instance, terms like \(4y \sqrt[3]{x}\) or \(3x^2 + 5x - 7\) are classic examples of algebraic expressions.
These expressions generalize arithmetic operations using variables, allowing us to formulate and solve problems with unknowns. Variables are symbols, often letters, that represent numbers in expressions or equations.
Transforming expressions into more workable forms is a big part of algebra, whether it's through expanding products, factoring, or rewriting roots as powers. Using positive rational exponents, like changing \(\sqrt[3]{x}\) to \(x^{\frac{1}{3}}\), is a key strategy. This form allows for more flexibility in applying algebraic rules like distributive, associative, and commutative properties, helping simplify the problem-solving process."