Problem 25 - Algebra 2 and Trigonometry

Question

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ f(x)=(2 x)^{-6} \div x^{3} ; f(-3) $$

Step-by-Step Solution

Answer

\(f(-3) = \frac{1}{-1259712}\).

Step 1: Substitute the Given Value of x

The function is given as \(f(x)=(2x)^{-6} \div x^{3}\). Substitute \(x = -3\) into the function to find \(f(-3)\). This becomes:\(f(-3) = (2(-3))^{-6} \div (-3)^{3}\).

Step 2: Simplify Inside the Parentheses for (2x)

Calculate \(2(-3)\) to simplify the expression:\(2(-3) = -6\).Now the function becomes:\(f(-3) = (-6)^{-6} \div (-3)^{3}\).

Step 3: Evaluate the Powers

Next, evaluate the powers:\((-6)^{-6} = \frac{1}{(-6)^{6}}\).Also, evaluate \((-3)^{3}\):\((-3)^{3} = -27\).Now the expression becomes:\(f(-3) = \frac{1}{(-6)^{6}} \div -27\).

Step 4: Calculate (-6)^6

Calculate \((-6)^{6}\) by multiplying \(-6\) by itself 6 times:\((-6)^{6} = 46656\).Therefore, \((-6)^{-6} = \frac{1}{46656}\).

Step 5: Simplify the Division

Substitute the evaluated powers into the expression:\(f(-3) = \frac{1}{46656} \div -27\).This is equivalent to:\(f(-3) = \frac{1}{46656} \times \frac{1}{-27}\).Multiply the fractions:\(f(-3) = \frac{1}{46656 \times -27}\).

Step 6: Calculate the Final Result

Multiply the denominators: \(46656 \times -27 = -1259712\).Thus, \(f(-3) = \frac{1}{-1259712}\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation, involving two numbers, the base and the exponent. The base is the number that is multiplied by itself as many times as indicated by the exponent. For example, in \(a^b\), \(a\) is the base, and \(b\) is the exponent.
- When an exponent is positive, you multiply the base by itself.- For instance, \(4^3 = 4 \times 4 \times 4 = 64\).
In our example, we dealt with \((-6)^{-6}\). First, we compute \((-6)^{6}\) by multiplying \(-6\) six times. This step results in 46,656 before dealing with the negative exponent.
Remember, understanding how exponents work allows you to simplify expressions that involve powers quickly. It's a fundamental skill in algebra, often encountered in formulas and geometric calculations.

Substitution

Substitution involves replacing a variable in a function or expression with a given value. In mathematics, it’s a key process used to simplify or solve equations.
When working with functions, substitution allows you to find the output, or value of the function, for given inputs.
In our exercise, we were given the function \(f(x)=(2x)^{-6} \div x^{3}\) and asked to evaluate it for \(x = -3\).
This step meant replacing every \(x\) in the function with the value -3. Thus, the function becomes \(f(-3) = (2(-3))^{-6} \div (-3)^3\).
- This laid the groundwork for simplifying the expression step by step.
By using substitution correctly, you can transform a general function into a more manageable expression that offers a specific numerical result.

Rational Expressions

Rational expressions are fractions that have polynomials in the numerator, the denominator, or both. Understanding how to manipulate these expressions is a vital part of algebra.
In this problem, after performing substitution and evaluating the powers, the expression simplified to a fraction: \(\frac{1}{(-6)^6} \div -27\).
- This involves dividing one rational expression by another.
Simplifying complex fractions usually requires multiplying by the reciprocal.In our case, we converted \(\frac{1}{(-6)^6} \div -27\) into \(\frac{1}{(-6)^6} \times \frac{1}{-27}\).
This method makes it easier to multiply the fractions across the numerators and denominators.
Working with rational expressions helps you develop skills for simplifying, adding, subtracting, multiplying, and dividing complex fractions.

Negative Exponents

Negative exponents represent reciprocal operations in mathematics. When the exponent of a number is negative, it indicates the reciprocal of the number raised to the corresponding positive exponent.
For example, \(a^{-b} = \frac{1}{a^b}\).In our function, we evaluated \((-6)^{-6}\), which translates to \(\frac{1}{(-6)^6}\).
- This switch from a negative exponent to a positive exponent's reciprocal is crucial for simplification.
Understanding negative exponents aids in solving equations more efficiently, especially when transforming intricate algebraic functions into simpler forms.
By demystifying negative exponents, you will boost your confidence in handling various mathematical expressions and problems.