Problem 90 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{(3 x+2)^{12}\left(\frac{1}{3}\right)(2 x+3)^{-23}(2)-(2 x+3)^{10}\left(\frac{1}{2}\right)(3 x+2)^{-12}(3)}{\left[(3 x+2)^{1 / 2}\right]^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the expression is by breaking down each term and combining exponents using exponent rules.

Step 1: Simplify the Expression in the Numerator

Start by simplifying the numerator: \[ (3x+2)^{12}\left(\frac{1}{3}\right)(2x+3)^{-23}(2) - (2x+3)^{10}\left(\frac{1}{2}\right)(3x+2)^{-12}(3) \]Distribute the constants for each term:- First term: \[ \frac{2}{3}(3x+2)^{12}(2x+3)^{-23} \]- Second term: \[ \frac{3}{2}(2x+3)^{10}(3x+2)^{-12} \]

Step 2: Apply Properties of Exponents

Apply the properties of exponents to combine powers of like bases for each expression:- For the first term, factor out similar bases:\[ \frac{2}{3} \cdot ((3x+2)^{12}(2x+3)^{-23}) \]- For the second term, factor out similar bases:\[ \frac{3}{2} \cdot ((2x+3)^{10}(3x+2)^{-12}) \]

Step 3: Rewrite Denominator Using Exponent Property

Rewrite the denominator using properties of exponents: \[ \left[(3x+2)^{1/2}\right]^2 = (3x+2)^{1/2 \times 2} = (3x+2)^{1} \]

Step 4: Combine and Simplify the Expressions

Combine the numerator terms:Combine like bases using the property \[ a^m a^n = a^{m+n} \]to simplify:- The numerator: \[ \frac{2}{3} \cdot (3x+2)^{12}(2x+3)^{-23} - \frac{3}{2} \cdot (2x+3)^{10}(3x+2)^{-12} \]- The denominator:\[ (3x+2)^1 \]

Step 5: Simplify Expression by Canceling Like Terms

Factor out the simplified terms, cancel the same terms in the numerator and denominator:- For the first fraction\[ \frac{(3x+2)^{12}}{(3x+2)^{1}} = (3x+2)^{11} \]- For the second fraction\[ \frac{(3x+2)^{-12}}{(3x+2)^{1}} = (3x+2)^{-13} \]Thus the numerator simplifies to:\[ \frac{2}{3} \cdot (3x+2)^{11}(2x+3)^{-23} - \frac{3}{2} \cdot (2x+3)^{10}(3x+2)^{-13} \]

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent. When we have an expression like \( (3x+2)^{12} \), the base is \( 3x+2 \) and the exponent is 12. This means \( 3x+2 \) is multiplied by itself 12 times. Exponentiation is a compact way to express repeated multiplication, and it is crucial for simplifying polynomial expressions in algebra.
To handle expressions involving exponents, always identify the base and the exponent. This allows you to apply various rules related to powers, making the expression much easier to manage as you work through it.

Properties of Exponents

Understanding the properties of exponents is key to simplifying expressions efficiently. These properties include product of powers, power of a power, and power of a quotient.

• Product of Powers: When multiplying like bases, add the exponents: \( a^m imes a^n = a^{m+n} \). This helps in combining terms like \( (3x+2)^{12} \times (3x+2)^{-1} \) into \( (3x+2)^{11} \).
• Power of a Power: When raising an exponent to another power, multiply the exponents: \( (a^m)^n = a^{m imes n} \). For instance, \( [(3x+2)^{1/2}]^2 \) simplifies to \( (3x+2)^{1} \).
• Power of a Quotient: When dividing like bases, subtract the exponents: \( a^m / a^n = a^{m-n} \). This rule is applied when simplifying fractions in expressions with exponents.

By mastering these properties, you will find it easier to handle more complex algebraic expressions.

Numerator and Denominator

In any fraction, the numerator is the top part and the denominator is the bottom part. Simplifying expressions involves working with both parts effectively.
The numerator in our original expression contains terms with different exponents that need to be handled carefully. Each term such as \( \frac{2}{3}(3x+2)^{12}(2x+3)^{-23} \) should be simplified separately before combining. In this step, distribute constants across the terms and utilize properties of exponents to simplify similar bases.
The denominator can sometimes be simplified using exponent rules as shown, where \( [(3x+2)^{1/2}]^2 \) reduces to \( (3x+2)^{1} \). Understanding this reduction helps to clarify operations like canceling out terms with similar bases, streamlining both the numerator and denominator efficiently.

Combining Like Terms

Combining like terms involves simplifying expressions where terms have the same variables raised to the same exponents. This concept is essential for both reducing expression complexity and finding a simplified version.
In a step-by-step solution, first identify terms with the same base, such as \( (3x+2) \) and \( (2x+3) \), and combine their exponents where possible. For example, using properties of exponents, \( (3x+2)^{12} \) and \( (3x+2)^{-1} \) can combine to \( (3x+2)^{11} \).
Besides simplifying within each part, evaluate subtraction or addition between similar terms carefully, and maintain a clear order. This not only reduces the expression but makes the final outcome far more manageable. Understanding how and when to properly combine like terms allows precise calculations and refines your problem-solving skills in algebra.