Problem 19 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$\left(-9^{3 / 4}\right)^{2}$$

Step-by-Step Solution

Answer

The expression evaluates to 27.

Step 1: Understand the Expression

We are asked to evaluate \((-9^{3/4})^2\). The expression inside the parentheses is a power of a negative number, but the exponent applies only to the base 9. The base 9 is raised to the power \(\frac{3}{4}\), then the result is raised to the power 2.

Step 2: Compute the Inner Exponent

Calculate \(9^{3/4}\). The exponent \(3/4\) means taking the fourth root of 9 and then cubing the result: \((9^{1/4})^3\). Since 9 is a positive number, we can compute \(9^{1/4} = \sqrt{\sqrt{9}} = \sqrt{3}\). Then cube the result: \((\sqrt{3})^3 = 3^{3/2}\).

Step 3: Apply the Outer Exponent

Now, take the result from Step 2, which is \((3^{3/2})^2\). Use the power of a power property of exponents by multiplying the exponents: \(3^{3/2 \times 2} = 3^3\).

Step 4: Compute the Final Result

Evaluate \(3^3\). This is simply \(3 \times 3 \times 3 = 27\). The negative sign in \((-9^{3/4})^2\) becomes positive because of the even power of 2, so the result remains positive.

Key Concepts

Power of a Power Property

When you encounter expressions like \((a^m)^n\), you're dealing with the power of a power property. This rule simplifies nested exponents by multiplying them together. This means \((a^m)^n = a^{m imes n}\).

For example, in the expression \((3^{3/2})^2\), you apply this property:

• Multiply the exponents: \(\frac{3}{2} \times 2 = 3\).
• The expression becomes \(3^3\).

Using the power of a power property helps to simplify calculations, making it easier to evaluate complex expressions efficiently.

Rational Exponents

Rational exponents like \(a^{m/n}\) combine roots and powers. The denominator represents the root, and the numerator represents the power. For instance, \(9^{3/4}\) involves taking the fourth root of 9 and cubing the result.

• First, find the fourth root: \(9^{1/4} = \sqrt{\sqrt{9}} = \sqrt{3}\).
• Then raise it to the third power: \((\sqrt{3})^3 = 3^{3/2}\).

This approach is handy when dealing with non-integer exponents, allowing you to break down complex expressions using simpler steps.

Exponent Rules

Exponent rules are fundamental to simplifying expressions and solving equations. They include rules for multiplying, dividing, and raising powers to powers. Key rules include:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(a^m / a^n = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)

These rules streamline calculations, providing shortcuts for handling complex expressions, as seen in solving \((-9^{3/4})^2\). Knowing when and how to apply each rule is crucial for mastering exponentiation.