Problem 51 - 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. $$\frac{(k+5)^{1 / 2}(k+5)^{-1 / 4}}{(k+5)^{3 / 4}}$$

Step-by-Step Solution

Answer

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

Step 1: Apply Exponent Rules in the Numerator

The expression in the numerator is \((k+5)^{1/2} \cdot (k+5)^{-1/4}\). Since the bases are the same, apply the rule of exponents that states \(a^m \cdot a^n = a^{m+n}\). So, \((k+5)^{1/2} \cdot (k+5)^{-1/4} = (k+5)^{1/2 - 1/4}\).

Step 2: Simplify the Exponent in the Numerator

Calculate the simplified exponent in the numerator: \(1/2 - 1/4 = 2/4 - 1/4 = 1/4\). Thus, the numerator simplifies to \((k+5)^{1/4}\).

Step 3: Apply Exponent Rules to the Entire Fraction

The whole fraction becomes \(\frac{(k+5)^{1/4}}{(k+5)^{3/4}}\). Using the rule \(\frac{a^m}{a^n} = a^{m-n}\), we get \((k+5)^{1/4 - 3/4}\).

Step 4: Simplify the Exponent in the Fraction

Subtract the exponents: \(1/4 - 3/4 = -2/4 = -1/2\). Thus, the fraction simplifies to \((k+5)^{-1/2}\).

Step 5: Write with Positive Exponents

To write the expression with a positive exponent, use the property that \(a^{-m} = \frac{1}{a^m}\). Therefore, \((k+5)^{-1/2} = \frac{1}{(k+5)^{1/2}}\).

Key Concepts

Exponent Rules

Exponent rules form a fundamental part of algebra, helping you simplify expressions involving powers. Here, we'll focus on the rules used in the original exercise.

You should remember the basic exponent rules:

• Product Rule: When multiplying two powers with the same base, add their exponents: \( a^m \cdot a^n = a^{m+n} \).
• Quotient Rule: When dividing two powers with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: To raise a power to another power, multiply the exponents: \( (a^m)^n = a^{mn} \).

Each of these rules relies on the consistency of the base, meaning they only apply when the bases are identical.
In our exercise, we applied the product rule and quotient rule to simplify the expression step by step. Understanding these rules can make dealing with exponential expressions far more manageable.

Positive Exponents

Positive exponents are straightforward and signify the number of times a number or variable is multiplied by itself. However, dealing with negative exponents requires a bit more attention.
Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. For example, \( a^{-m} = \frac{1}{a^m} \).

The exercise instructs you to convert expressions with negative exponents into those with positive exponents. This step can be essential for interpreting results clearly and aligns with standard practices in mathematics and science. In our solution, we converted \((k+5)^{-1/2}\) into the expression \(\frac{1}{(k+5)^{1/2}}\) to achieve a positive exponent. Always remember, positive exponents make expressions simpler and more intuitive.

Algebraic Fractions

Algebraic fractions function similarly to standard fractions but involve variables. Simplifying these fractions often necessitates both understanding basic algebraic operations and exponent rules.

When working with algebraic fractions, consider:

• Finding common bases to combine powers.
• Simplifying by reducing the expression through exponent subtraction or addition.
• Rewriting expressions to make them easier to interpret.

This exercise featured an algebraic fraction, \( \frac{(k+5)^{1/4}}{(k+5)^{3/4}} \), where exponents were simplified using division or subtraction of exponents as per the quotient rule. This simplification allowed a clearer final result, converting a complex, bulky expression into a more streamlined and manageable form. Mastery of algebraic fractions enables you to tackle a wide array of math problems efficiently.