Problem 82 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\left(x^{2}+9\right)^{4}\left(-\frac{1}{3}\right)(x+6)^{-43}+(x+6)^{-10}(4)\left(x^{2}+9\right)^{3}(2 x)$$

Step-by-Step Solution

Answer

The expression simplifies to: \[-\frac{1}{3}(x^2+9)^4(x+6)^{-43} + 8x(x+6)^{-10}(x^2+9)^3\]

Step 1: Simplify Each Term Individually

First, let's look at the expression: \((x^2+9)^4\left(-\frac{1}{3}\right)(x+6)^{-43}\) and \((x+6)^{-10}(4)(x^2+9)^3(2x)\). Notice that each term is a product of several factors.

Step 2: Analyze and Simplify the Coefficients

The first term: - The coefficient is \(-\frac{1}{3}\).The second term:- The coefficient is \(4\).- The \(2x\) contributes an additional \(2\), so the combined coefficient for the second term is \(8x\).

Step 3: Apply Laws of Exponents

The first term is already simplified: - The terms \((x^2+9)^4\) and \((x+6)^{-43}\) remain as they are. For the second term, we notice:- We have \((x+6)^{-10}\) and \((x^2+9)^3\). - Combine the exponents where possible, but here there are no common bases, so we just rewrite them.

Step 4: Combine the Terms

Since the terms do not have the same bases and exponents, the terms cannot be directly combined.- The expression remains as is after coefficients are simplified: \[-\frac{1}{3}(x^2+9)^4(x+6)^{-43} + 8x(x+6)^{-10}(x^2+9)^3\]

Step 5: Final Simplified Expression

The simplified expression from Step 4 is already as simple as it can get given the different bases and powers.Hence, the final expression is: \[-\frac{1}{3}(x^2+9)^4(x+6)^{-43} + 8x(x+6)^{-10}(x^2+9)^3\]

Key Concepts

Laws of Exponents

When working with algebraic expressions involving exponents, it's essential to master the Laws of Exponents. These rules help simplify expressions that contain powers.

• **Product of Powers Rule**: If you multiply two powers with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
• **Quotient of Powers Rule**: If you divide two powers with the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\).
• **Power of a Power Rule**: If you raise a power to another power, multiply the exponents: \((a^m)^n = a^{mn}\).
• **Power of a Product Rule**: If you want to raise a product to a power, distribute that power to each factor: \((ab)^n = a^n b^n\).
• **Zero Exponent Rule**: Any non-zero number raised to the power of zero equals 1: \(a^0= 1\).
• **Negative Exponent Rule**: A negative exponent indicates that the base is on the wrong side of the fraction line, so you flip it: \(a^{-n} = \frac{1}{a^n}\).

These rules are powerful tools for simplifying expressions accurately. By carefully applying them, you can handle complex algebraic expressions with ease.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations (such as addition or multiplication). These are the building blocks of algebra and come in various forms, such as polynomials, rational expressions, and others. Here are some important components:

• **Terms**: The parts of an expression separated by addition or subtraction. For example, in \(3x + 4y - 5\), \(3x\), \(4y\), and \(-5\) are terms.
• **Coefficients**: Numbers multiplying the variables, such as \(3\) in \(3x\).
• **Constants**: Numbers without variables, like \(-5\) in \(3x + 4y - 5\).
• **Variables**: Symbols representing numbers, \(x\) or \(y\), which can take various values.

Understanding these components helps you see how expressions are structured and interact when combined or simplified. Expressions can range from very simple, with only a few terms, to complex, with multiple terms and operations nested within each other.

Expression Simplification

Simplifying algebraic expressions involves reducing them into the simplest form possible. It's about achieving clarity and compactness without altering the original meaning or value. Here are some steps and tips:

• **Combine like terms**: These are terms with the same variables raised to the same power. For instance, in \(2x + 3x\), both terms can combine to make \(5x\).
• **Use the distributive property**: This rule states \(a(b+c) = ab + ac\). It allows you to expand or factor expressions, which helps in simplification.
• **Factor when possible**: Look for common factors in terms. For example, in \(x^2 - 2x\), you can factor out \(x\) resulting in \(x(x-2)\).
• **Apply all applicable Laws of Exponents**: This reduces exponents wherever possible for more straightforward expressions.

The goal of simplification is to make an expression easier to work with, whether solving an equation or evaluating it for specific variable values. By simplifying expressions, you can better understand the relationships between different parts of the equation or problem.