Question

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$6(6 x+1)^{1 / 3}(4 x-3)^{3 / 2}+6(6 x+1)^{4 / 3}(4 x-3)^{1 / 2} \quad x \geq \frac{3}{4}$$

Short answer

The factored expression is \[6(6 x+1)^{1 / 3}(4 x-3)^{1 / 2}(10 x-2).\]

Step-by-step solution

Identify the common factor

First, identify the common factors in both terms. Notice that both terms share the factor 6, \(6x+1\)^{1/3}, and \(4x-3\)^{1/2}.

Factor out the common factor

Factor out the common factor from the expression: \(6 \(6x+1\)^{1/3}\(4x-3\)^{1/2}\).

Simplify the remaining terms

Rewrite the expression with the factored out part: \[6(6 x+1)^{1 / 3}(4 x-3)^{3 / 2}+6(6 x+1)^{4 / 3}(4 x-3)^{1 / 2}\ = 6(6 x+1)^{1 / 3}(4 x-3)^{1 / 2} \left[(4 x-3)+ (6 x+1) \right] \]

Combine and simplify the parentheses

Add the expressions inside the parentheses: \[6(6 x+1)^{1 / 3}(4 x-3)^{1 / 2} (4x - 3 + 6x + 1).\] Simplify the terms inside: \[4x - 3 + 6x + 1 = 10x - 2\]

Write the final expression

Combine the factors and the simplified term to get the final expression: \[6(6 x+1)^{1 / 3}(4 x-3)^{1 / 2}(10 x-2). \]

Key concepts

Exponents Simplification

When dealing with algebraic expressions, you often encounter exponents. Simplifying exponents is crucial for making calculations easier and for better understanding.
Simplification involves making the exponent values more manageable, usually by finding a common term.
For example, in the given problem, we had exponents of \( \frac{1}{3} \) and \( \frac{3}{2} \). The idea was to express the terms in such a way that the exponents become positive and manageable.
This step ensures clarity and streamlines further calculations. Remember, exponential rules like \(a^m \cdot a^n = a^{m+n} \) and \( (a^m)^n = a^{mn} \) come in handy.

Common Factor

Finding a common factor is like finding a common theme in a story. It's something that multiple terms in an equation share.
In our case, the exercise required identifying common factors in both terms of the given expression. By examining \(6(6 x+1)^{1/3}(4 x-3)^{3 / 2}+6(6 x+1)^{4 / 3}(4x-3)^{1/2} \), we found that both terms had a common factor of \(6, (6x+1)^{1/3} \) and \( (4x-3)^{1/2} \).
By factoring out these common elements, the complex expression becomes simpler.
This process helps in reducing the algebraic expression to a more manageable form, making it easier to solve or further manipulate.

Polynomial Addition

Polynomials are expressions involving variables raised to different powers added together. Adding polynomials involves combining like terms, which are terms with the same variable and the same exponent.
In the exercise, after factoring out the common terms, we had \[6(6 x+1)^{1 / 3}(4 x-3)^{1 / 2} \left[(4 x-3)+ (6 x+1) \right]\] This required adding \(4x-3 \) and \(6x+1 \).
By combining these like terms (the x terms and constant terms), we get \(10x - 2 \). Now, the polynomial addition has made the expression simpler and ready for the final multiplication.
Understanding polynomial addition is important for mastering more advanced algebraic techniques.