Problem 34 - Precalculus Mathematics for Calculus

Question

Simplify the expression. $$\sqrt[3]{2 y^{4}}-\sqrt[3]{y}$$

Step-by-Step Solution

Answer

The simplified expression is \( y^{\frac{1}{3}}(2^{\frac{1}{3}}y^{\frac{1}{3}} - 1) \).

Step 1: Express in terms of powers

First, convert the cube root to an expression with exponents. The cube root of any term can be expressed as raising to the power of \( \frac{1}{3} \). Here, we rewrite \( \sqrt[3]{2} \) as \( 2^{\frac{1}{3}} \) and \( \sqrt[3]{y} \) as \( y^{\frac{1}{3}} \). The expression becomes \( (2y^4)^{\frac{1}{3}} - y^{\frac{1}{3}} \).

Step 2: Apply exponent rules

Apply the rules of exponents to simplify \( (2y^4)^{\frac{1}{3}} \). We use the power of a product rule, which says \( (ab)^n = a^n b^n \), to get \( 2^{\frac{1}{3}}y^{\frac{4}{3}} \). The expression is now \( 2^{\frac{1}{3}}y^{\frac{4}{3}} - y^{\frac{1}{3}} \).

Step 3: Factor the expression

Recognize that both terms contain a common factor of \( y^{\frac{1}{3}} \). Factor \( y^{\frac{1}{3}} \) out of each term in the expression. You obtain \( y^{\frac{1}{3}}(2^{\frac{1}{3}}y^{\frac{1}{3}} - 1) \).

Step 4: Final Simplification

After factoring, check if any further simplification is possible. There are no like terms left to combine or further factors to apply, so the expression is now fully simplified.

Key Concepts

Exponent Rules

Exponent rules are a set of guidelines for simplifying mathematical expressions involving powers. They help us manage complex expressions by reducing them step-by-step using various rules. One important rule is the 'power of a product' rule. This rule states that to raise a product to a power, you raise each factor in the product to that power separately. For instance, if you have \[(ab)^n = a^n \times b^n\]This rule allows us to work with each part of a product individually. In our exercise, we used this rule to simplify the expression \[(2y^4)^{\frac{1}{3}}\]into\[2^{\frac{1}{3}}y^{\frac{4}{3}}\].
Other common exponent rules include:

• The 'power of a power' rule: \((a^m)^n = a^{m\times n}\)
• The 'product of powers' rule: \(a^m \times a^n = a^{m+n}\)

Understanding these rules is essential for simplifying expressions with exponents effectively.

Cube Roots

Cube roots are the opposite of cubing a number. Finding a cube root means identifying a number which, when used in three equal factors, results in the original number. For any number or expression \(x\), the cube root is represented as \(\sqrt[3]{x}\), which can also be written in exponential form as \(x^{\frac{1}{3}}\).
In the exercise, the initial expression involved cube roots, suggested by the symbol \(\sqrt[3]{}\). When we rewrite cube roots in exponential notation, it allows us to use exponent rules to simplify the expressions further. Converting cube roots to exponents can make it easier to conduct operations like addition, subtraction, and factoring. This is crucial when simplifying complex expressions, as seen in how we converted \(\sqrt[3]{2y^4}\) to \[(2y^4)^{\frac{1}{3}}\].By applying exponent rules, cube roots become manageable in various types of algebraic expressions.

Factoring

Factoring is a technique used to break down expressions into simpler 'factorable' parts or products of their factors. It's akin to "reverse engineering" a multiplication problem. In algebra, factoring is a key skill for simplifying expressions, solving equations, and finding the greatest common divisor. The goal of factoring is to see if an expression can be rewritten as a product of simpler expressions. In our example, the expression \[2^{\frac{1}{3}}y^{\frac{4}{3}} - y^{\frac{1}{3}}\]contains a common factor of \(y^{\frac{1}{3}}\). By factoring \(y^{\frac{1}{3}}\) out, you simplify the expression to\[y^{\frac{1}{3}}(2^{\frac{1}{3}}y^{\frac{1}{3}} - 1)\].
This simplification helps us identify expressions that cannot be reduced further or combined with like terms. Factoring is a precious tool in algebra for tidying up expressions and equipping you with a clearer view of equation solutions.