Problem 36 - College Algebra

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. $$\left(\frac{a}{2^{3}}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{a^2}{64}\).

Step 1: Apply the Power of a Quotient Rule

The expression given is \( \left(\frac{a}{2^{3}}\right)^{2} \). According to the power of a quotient rule, \( \left(\frac{x}{y}\right)^{n} = \frac{x^{n}}{y^{n}} \). Apply this rule to the expression: \( \frac{a^{2}}{(2^{3})^{2}} \).

Step 2: Simplify the Denominator

Simplify \((2^3)^2\) in the denominator using the power of a power property, which is \((x^m)^n = x^{m \cdot n}\). This gives \(2^{3 \cdot 2} = 2^{6}\). Hence, the expression becomes \(\frac{a^{2}}{2^{6}}\).

Step 3: Ensure Positive Exponents

Check the expression \(\frac{a^2}{2^6}\). Since all exponents in the expression are already positive, no further changes are necessary.

Key Concepts

Power of a Quotient

The concept of "Power of a Quotient" is a fundamental rule when working with algebraic expressions involving fractions. It simplifies the process of dealing with exponents in fractions. When an expression like \( \left(\frac{x}{y}\right)^{n} \) is raised to a power, the power applies to both the numerator \(x\) and the denominator \(y\) separately. This results in \( \frac{x^{n}}{y^{n}} \).
For example, when simplifying an expression like \( \left(\frac{a}{2^3}\right)^2 \), we apply the power of a quotient rule:

• The power 2 applies to the numerator \(a\), which becomes \(a^2\)
• It also applies to the denominator \(2^3\), turning it into \((2^3)^2\).

This distributes the exponent across the fraction, making it easier to simplify.

Exponents

Understanding how exponents work is crucial for simplifying expressions efficiently. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, in \(2^3\), 2 is the base and 3 is the exponent. This tells us to multiply 2 by itself three times, i.e., \(2 \times 2 \times 2 = 8\).
When we encounter exponents raised to other exponents, like in \((x^m)^n\), we use the power of a power rule. This rule states that you multiply the exponents: \((x^m)^n = x^{m \cdot n}\).
In our original exercise, simplifying the denominator \((2^3)^2\) involves applying the power of a power rule to get \(2^{3 \cdot 2} = 2^6\). This step is essential for handling complex expressions and ensuring calculations are correct.

Simplifying Expressions

The process of simplifying expressions focuses on reducing them to their simplest form while keeping operations and mathematical rules intact. Simplification generally involves using various algebraic rules including those of exponents, distributing powers, and combining like terms.

• When simplifying, we apply rules like the power of a quotient to distribute exponents over a fraction.
• We then reduce any part of the expression that can be simplified.

For the expression \(\frac{a^2}{2^6}\), we checked the exponents and confirmed they were positive as a final step. If we had negative exponents, we'd rearrange terms or further simplify to ensure all exponents stay positive. Simplified expression forms are easier to interpret and are often required in final answers in algebra.