Problem 559 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(64^{\frac{5}{2}}\) (b) \(81^{-\frac{3}{2}} (c) 27^{-\frac{4}{3}}\)

Step-by-Step Solution

Answer

64^{\frac{5}{2}} = 32768, \, 81^{-\frac{3}{2}} = \frac{1}{729}, \, 27^{-\frac{4}{3}} = \frac{1}{81}

Step 1: Simplify Exponent Expression for (a)

For the expression \(64^{\frac{5}{2}}\), recognize that 64 can be written as \(8^2\)\[64 = 8^2\]. Thus, the expression becomes \( (8^2)^{\frac{5}{2}} \). Using the power rule \((a^m)^n = a^{mn}\), simplify to: \[ (8^2)^{\frac{5}{2}} = 8^{2 \cdot \frac{5}{2}} = 8^5 = 32768\].

Step 2: Simplify Exponent Expression for (b)

For the expression \(81^{-\frac{3}{2}} \), recognize that 81 can be written as \(9^2\)\[ 81 = 9^2 \]. Thus, the expression becomes \((9^2)^{-\frac{3}{2}} \). Using the power rule \((a^m)^n = a^{mn}\), simplify to:\[ (9^2)^{-\frac{3}{2}} = 9^{2 \cdot -\frac{3}{2}} = 9^{-3} = \frac{1}{9^3} = \frac{1}{729} \].

Step 3: Simplify Exponent Expression for (c)

For the expression \(27^{-\frac{4}{3}}\), recognize that 27 can be written as \(3^3\)\[ 27 = 3^3 \]. Thus, the expression becomes \((3^3)^{-\frac{4}{3}} \). Using the power rule \((a^m)^n = a^{mn} \), simplify to:\[ (3^3)^{-\frac{4}{3}} = 3^{3 \cdot -\frac{4}{3}} = 3^{-4} = \frac{1}{3^4} = \frac{1}{81} \].

Key Concepts

Exponent Rules

Exponent rules are essential in simplifying expressions involving powers. An exponent signifies how many times a base number is multiplied by itself. For instance, in the expression \(a^m\), \(a\) is the base, and \(m\) is the exponent.
These rules help in performing various operations with exponents comfortably.
Most common ones include:

• Product Rule: When multiplying like bases, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Quotient Rule: When dividing like bases, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power Rule: When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{mn}\).
• Zero Exponent Rule: Any base raised to the power of zero is 1: \(a^0 = 1\), provided \(a eq 0\).

Understanding these rules allows you to simplify complicated expressions and solve exponential equations effectively.

Rational Exponents

Rational exponents are exponents that are fractions. They bring together exponentiation and roots in a single notation.
For example, the expression \(a^{\frac{m}{n}}\) can be interpreted in two ways:

• \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)
• \(a^{\frac{m}{n}} = (\sqrt[n]{a})^m\)

Both notations are numerically equivalent. With rational exponents, simplification often involves recognizing patterns in the bases, simplifying roots, and applying the power rule.
In the given exercise, we simplified \(64^{\frac{5}{2}}\) by rewriting 64 as \(8^2\) and applying the power rule: \((8^2)^{\frac{5}{2}} = 8^5\).

Negative Exponents

Negative exponents indicate division or reciprocals. The general rule is:

• \(a^{-m} = \frac{1}{a^m}\)

This helps transform a power with a negative exponent into its reciprocal with a positive exponent.
For example, in the solution involving \(81^{-\frac{3}{2}}\), we used this rule as follows: \(9^{-3} = \frac{1}{9^3} = \frac{1}{729}\).
Understanding negative exponents is crucial for simplifying and evaluating exponential expressions that initially seem complex.

Power Rule

The Power Rule states that when raising a power to another power, you multiply the exponents: \((a^m)^n = a^{mn}\). This rule simplifies expressions where the base is raised to multiple exponents.
For instance, in the exercise solution for \(27^{-\frac{4}{3}}\), we recognized that 27 is \(3^3\). Using the Power Rule, we transformed \((3^3)^{-\frac{4}{3}}\) into \(3^{3 \cdot -\frac{4}{3}} = 3^{-4}\).
Ultimately, simplifying it further as a fraction gave us \(\frac{1}{3^4} = \frac{1}{81}\).
The Power Rule is pivotal when simplifying nested exponents in expressions, making it a fundamental tool in your math toolbox.