Problem 80 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$(3 x+2)^{1 / 3}(2)(4 x-5)(4)+(4 x-5)^{2}(3)(3 x+2)^{-2 / 3}(3)$$

Step-by-Step Solution

Answer

The simplified expression is \(17(4x-5)^3(3x+2)^{-1/3}\).

Step 1: Distribute constants

First, let's distribute the constants across each term. We have:\((3x+2)^{1/3} \times (2) \times (4x-5) \times (4) = 8(3x+2)^{1/3}(4x-5)\) for the first term, and \((4x-5)^{2} \times (3) \times (3x+2)^{-2/3} \times (3) = 9(4x-5)^{2}(3x+2)^{-2/3}\) for the second term.

Step 2: Combine like terms

Notice that both terms have parts involving \((3x+2)\) and \((4x-5)\). We can express the first term as \((3x+2)^{1/3} \cdot (4x-5)^1\) and the second term as \((4x-5)^{2} \cdot (3x+2)^{-2/3}\).

Step 3: Combine bases with exponents

Group the terms with the same bases:- For \((3x+2)\), we have powers \(1/3\) and \(-2/3\).- For \((4x-5)\), we have powers \(1\) and \(2\).Combine the exponents for each base:\((3x+2)^{1/3 - 2/3} = (3x+2)^{-1/3}\)\((4x-5)^{1+2} = (4x-5)^{3}\)

Step 4: Simplify and combine common factors

After combing the bases, our expression becomes:\[8(4x-5)^3 (3x+2)^{-1/3} + 9(4x-5)^3 (3x+2)^{-1/3}\]We see they both share a common factor of \((4x-5)^3 (3x+2)^{-1/3}\).Factor out \((4x-5)^3 (3x+2)^{-1/3}\), giving:\[(8 + 9)(4x-5)^3 (3x+2)^{-1/3} = 17(4x-5)^3 (3x+2)^{-1/3}\]

Step 5: Present the simplified expression

The fully simplified expression is \[17(4x-5)^3(3x+2)^{-1/3}\]. This is the most simplified form of the given expression using the steps of combining and factoring out common terms.

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when simplifying expressions with powers. These rules help us manage and simplify expressions, especially when dealing with multiplication or division involving the same base. Here are some key exponent rules:

• Product of Powers Rule: When multiplying two expressions with the same base, you add their exponents. For example, if you have a base \(b\), then \(b^m \times b^n = b^{m+n}\).
• Quotient of Powers Rule: When dividing two expressions with the same base, subtract the exponents. This can be expressed as \(b^m / b^n = b^{m-n}\).
• Negative Exponent Rule: A negative exponent indicates a reciprocal. For instance, \(b^{-n} = 1/b^n\).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \((b^m)^n = b^{m \times n}\).

Mastering these rules aids in combining and simplifying complex expressions, as seen in our given exercise involving terms like \((3x+2)^{1/3}\) and \((3x+2)^{-2/3}\). Recognizing that we can combine these using the product of powers rule allows us to find a simpler result.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operators. Simplifying these can often involve combining like terms or applying certain mathematical properties. In the context of the exercise, recognizing parts of the expression that can be combined simplifies the problem considerably.

An algebraic expression might include:

• Constants: Numbers without variables, such as 2, 3, or 4 in the exercise.
• Variables: Letters representing numbers, like \(x\).
• Terms: Parts of the expression separated by + or - signs. For example, \((3x+2)^{1/3}\) and \((4x-5)^{2}\).
• Like Terms: Terms with the same variables raised to the same powers. For instance, terms with \((4x-5)\) can be combined in our solution.

Identifying like terms and using properties of exponents helps in bringing together parts of an expression. This process not only simplifies but can also provide a more understandable form for further mathematical operations.

Factoring Expressions

Factoring involves breaking down an expression into simpler components—factors—that multiply together to make the original expression. In the context of simplifying expressions in algebra, factoring is a valuable tool for revealing common factors that can be simplified further.

To factor expressions, you can:

• Find common factors: Look for terms that appear in multiple parts of the expression. These can be factored out and simplified. In our solution, we found a common factor \((4x-5)^3 (3x+2)^{-1/3}\).
• Use special patterns: Recognize patterns like difference of squares or perfect square trinomials.
• Apply distributive property: Reverse the distributive property to factor out common terms. For example, if you have \(ax + ay\), it can be factored to \(a(x + y)\).

Factoring is useful not just for simplification, but also for solving equations. By simplifying down to key components, you can more easily analyze or solve for variables, making complex algebra more manageable and straightforward.