Problem 26 - Algebra 2 and Trigonometry

Question

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=\left(x^{-7}\right)^{4} ; \mathrm{f}(-6) $$

Step-by-Step Solution

Answer

\( f(-6) = \frac{1}{(-6)^{28}}.\)

Step 1: Understand the Function

The function given is \( f(x) = (x^{-7})^4 \). This means that for a particular value of \( x \), we first raise \( x \) to the \(-7\) power and then raise the result to the \(4\)th power.

Step 2: Substitute the Given Value

Substitute \( x = -6 \) into the function: \( f(-6) = ((-6)^{-7})^4 \).

Step 3: Simplify the Inner Exponentiation

Calculate \((-6)^{-7}\). The expression \( a^{-b} \) is equivalent to \( \frac{1}{a^b} \). So, \((-6)^{-7} = \frac{1}{(-6)^7} \).

Step 4: Simplify Entire Expression

Substitute back: \( f(-6) = \left( \frac{1}{(-6)^7} \right)^4 = \frac{1}{((-6)^7)^4} \). Identify that \((a^b)^c = a^{b \cdot c}\), so \((-6)^{-7})^4 = (-6)^{-28}.\)

Step 5: Final Evaluation

Calculate \((-6)^{-28}\). This simplifies to \(\frac{1}{(-6)^{28}}\). Since we are not required to evaluate \((-6)^{28}\) in complete numerical detail for this level, recognize it as a very small positive number.

Key Concepts

Negative Exponents

Understanding negative exponents is crucial in simplifying expressions like \( (-6)^{-7} \). A negative exponent tells us to take the reciprocal of the base raised to the opposite positive exponent. For example, \( x^{-n} = \frac{1}{x^n} \), where \( x \) is the base, and \( n \) is the exponent.

• This means that instead of multiplying \( x \) by itself \( n \) times, we divide \( 1 \) by \( x^n \).
• In our example, \( (-6)^{-7} \) is equivalent to \( \frac{1}{(-6)^7} \).
• This conversion transforms a seemingly complicated expression into a simpler fraction.

Recognizing and applying the negative exponent rule can dramatically simplify your calculations and help uncover the fundamental structure behind complex expressions.

Order of Operations

When evaluating functions and expressions, especially those involving exponents, the order of operations becomes crucial. You may have heard of PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Following this order ensures we evaluate expressions correctly.

• First, handle anything in parentheses. In \( ((-6)^{-7})^4 \), simplify \( (-6)^{-7} \) first.
• Next, address exponents. Here, once the expression inside the parentheses is simplified, you deal with \( ((-6)^{-7})^4 \).
• By respecting this order, you avoid common pitfalls in calculations, especially when dealing with nested exponents or multiple operations.

Practicing the order of operations can significantly improve your problem-solving efficiency and accuracy.

Power of a Power Rule

The power of a power rule is a handy tool when dealing with expressions where an exponent is raised to another exponent. This rule states that \( (a^b)^c = a^{b \cdot c} \).

• In the given exercise, we have \( ((-6)^{-7})^4 \). By applying the power of a power rule, this becomes \( (-6)^{-7 \cdot 4} \) or \( (-6)^{-28} \).
• It simplifies expressions quickly and reduces the chances of error.
• Understanding this rule is especially useful in higher-level mathematics where functions and equations become increasingly complex.

Thus, by employing the power of a power rule, solving equations with multiple layers of exponents becomes far more manageable and straightforward.