Problem 44 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$\frac{6 k^{-4}\left(3 k^{-1}\right)^{-2}}{2^{3} k^{1 / 2}}$$

Step-by-Step Solution

Answer

\(\frac{1}{12k^{5/2}}\)

Step 1: Simplify Inside Parentheses

Simplify the term \((3k^{-1})^{-2}\). First, raise both the coefficient and the variable to the power of -2: \((3^{-2})(k^{2})\) is equal to \(\frac{1}{9}k^{2}\).

Step 2: Simplify Numerator

Multiply the simplified result from Step 1 with \(6k^{-4}\): \(6k^{-4} \times \frac{1}{9}k^{2} = \frac{6}{9}k^{-4+2} = \frac{2}{3}k^{-2}\).

Step 3: Simplify Denominator

Simplify the denominator \(2^3k^{1/2}\) to \(8k^{1/2}\).

Step 4: Divide the Fractions

Divide the results from Step 2 by Step 3: \(\frac{\frac{2}{3}k^{-2}}{8k^{1/2}} = \frac{2}{3} \times \frac{1}{8}k^{-2-1/2}\). Simplify exponent by combining: \(k^{-2-1/2} = k^{-5/2}\).

Step 5: Simplify the Expression

Multiply the fractions: \(\frac{2}{3} \times \frac{1}{8} = \frac{2}{24} = \frac{1}{12}\). So, \(\frac{1}{12}k^{-5/2} = \frac{1}{12} \times \frac{1}{k^{5/2}} = \frac{1}{12k^{5/2}}\).

Key Concepts

Exponents

Exponents are a way to represent how many times a number, known as the base, is multiplied by itself. In the expression \( k^3 \), the number \( k \) is used as a factor three times. Exponents are incredibly useful for simplifying expressions, especially when dealing with large numbers or variables raised to powers.

When an exponent is negative, such as in \( k^{-2} \), it denotes the reciprocal of the base raised to the positive counterpart of the exponent: \( k^{-2} = \frac{1}{k^2} \). This allows you to transform expressions with negative exponents into fractions with positive exponents, making them simpler to work with.

Also, there are several key rules for working with exponents:

• Product Rule: \( a^m \times a^n = a^{m+n} \)
• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power Rule: \( (a^m)^n = a^{m \cdot n} \)

Using these rules helps in breaking down complex expressions and in the simplification process.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while keeping the equivalence intact. It is a silent art of looking for common terms and applying arithmetic operations along with algebraic rules to achieve a sleeker, more concise version of the original expression.

Here are steps generally involved in simplifying expressions:

• First, accomplish all operations inside any parentheses.
• Apply the distributive property where necessary, which states that \( a(b+c) = ab + ac \).
• Combine like terms: terms that have the exact same variables and powers, like \( 3x \) and \( 5x \), can be added or subtracted directly.
• Simplify any coefficients – fractions can be reduced by their greatest common factors.
• Finally, ensure that exponents are positive as many standard forms require, like the one in the given problem, signified by transforming \( x^{-n} \) to \( \frac{1}{x^n} \).

With practice, these steps become habitual and speed up the process of solving equations or evaluating complex expressions.

Fraction Operations

Fractions are expressions that represent a division of one quantity by another. They consist of a numerator and a denominator. In algebra, working with fractions often requires multiplying, dividing, and simplifying fractions which include algebraic expressions.

Multiplying fractions is straightforward: multiply the numerators together and the denominators together. Simplify wherever possible before multiplying to make the operation easier and to avoid large numbers. For example, \( \frac{2}{3} \times \frac{1}{8} = \frac{2 \times 1}{3 \times 8} = \frac{2}{24} = \frac{1}{12} \).

Dividing fractions requires multiplying by the reciprocal. For example, \( \frac{2}{3} \div \frac{1}{4} \) turns into \( \frac{2}{3} \times \frac{4}{1} \).

When combining fractions that have variables, pay close attention to the laws of exponents and operations. The same exponent rules apply — ensure positive exponents in the final simplified form by rewriting negative exponents as reciprocals. Also, simplify the fractions as much as possible to achieve the cleanest result every time.