Problem 58 - Intermediate Algebra

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(m^{2} n\right)^{1 / 4}}{m^{-1 / 2} n^{5 / 8}} $$

Step-by-Step Solution

Answer

\( \frac{m}{n^{3/8}} \)

Step 1: Distribute the Exponent

First, apply the exponent \( \frac{1}{4} \) to both \( m^2 \) and \( n \) inside the expression \( (m^2 n)^{1/4} \). Using the power of a power property, we have:\[ (m^2)^{1/4} = m^{2 \times \frac{1}{4}} = m^{1/2} \]\[ (n)^{1/4} = n^{1/4} \]So the expression becomes \( \frac{m^{1/2} n^{1/4}}{m^{-1/2} n^{5/8}} \).

Step 2: Simplify the Base with Exponents

Now simplify the expression using the quotient of powers property, which states \( \frac{a^m}{a^n} = a^{m-n} \).For the base \( m \), simplify:\[ \frac{m^{1/2}}{m^{-1/2}} = m^{1/2 - (-1/2)} = m^{1/2 + 1/2} = m^{1} \]For the base \( n \), simplify:\[ \frac{n^{1/4}}{n^{5/8}} = n^{1/4 - 5/8} \]Convert \( 1/4 \) to an equivalent fraction with the same denominator:\[ n^{2/8} \] So:\[ n^{2/8 - 5/8} = n^{-3/8} \]

Step 3: Final Expression with Positive Exponents

Ensure all exponents are positive. Since \( n^{-3/8} \) has a negative exponent, write it with a positive exponent by moving it to the denominator:\[ m \cdot n^{-3/8} = \frac{m}{n^{3/8}} \]Thus, the final simplified expression is:\[ \frac{m}{n^{3/8}} \].

Key Concepts

Properties of Exponents

Exponents are a shorthand way to express repeated multiplication. Understanding the properties of exponents helps in simplifying algebraic expressions. Let's look at the most commonly used properties:

• Product of Powers: When multiplying like bases, you add the exponents: if you have \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers: When dividing like bases, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: When raising a power to another power, you multiply the exponents: \( (a^m)^n = a^{m \times n} \).
• Zero Exponent: Any non-zero base raised to the zero power is one: \( a^0 = 1 \).
• Negative Exponents: A negative exponent indicates a reciprocal: \( a^{-m} = \frac{1}{a^m} \).

Each of these helps in manipulating and transforming expressions to simplify them, as seen in the original exercise.

Simplifying Algebraic Expressions

When simplifying algebraic expressions, particularly those containing exponents, we focus on applying the rules of exponents effectively. In the original problem, we started by distributing the exponent of \( \frac{1}{4} \) over each term inside the parentheses. This is an example of the power of a power property.
After simplifying these initial expressions, different bases and operations within the expression were tackled individually. Using the quotient of powers property helped simplify complex fractions within the expression.
By systematically applying properties of exponents and performing arithmetic on the exponents, algebraic expressions become simpler and more concise. This streamlines calculations and often reveals the solution in a more digestible form.

Positive Exponents

In algebra, expressing exponents positively simplifies understanding and usage in equations. Negative exponents represent fractions, indicating reciprocal values, and can sometimes complicate an expression.
Transforming negative exponents into positive ones requires placing them in the denominator or altering their positions within the expression. In the simplest terms, switch bases with negative exponents from the numerator to the denominator, or vice versa, to make them positive.
This is what was done in the final step of the exercise, converting \( n^{-3/8} \) to \( \frac{1}{n^{3/8}} \). This maneuver ensures that exponents remain non-negative, reducing complexity in algebraic solutions and making expressions more aesthetically pleasing and easier to handle.