Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-10: Express the number in the form \(a / b\), where \(a\) and \(b\) are integers. $$ 9^{5 / 2} $$

Step-by-Step Solution

Answer

The number is \(\frac{243}{1}\).

Step 1: Understand the Exponential Notation

The given expression is \(9^{5/2}\). This means that the base, 9, is raised to the power of 5/2.

Step 2: Rewrite the Base in Terms of a Square

Recognize that 9 can be expressed as a power: \(9 = 3^2\). Thus, \((3^2)^{5/2}\) is equivalent to \(9^{5/2}\).

Step 3: Apply the Power of a Power Rule

Apply the rule \((a^m)^n = a^{m \cdot n}\). In this case, \((3^2)^{5/2} = 3^{2 \cdot (5/2)} = 3^5\).

Step 4: Simplify 3 to the Power 5

Compute \(3^5\) by multiplying 3 by itself five times: \(3 \times 3 \times 3 \times 3 \times 3 = 243\).

Step 5: Express as a Fraction

Since \(3^5 = 243\), express this as a fraction. In this context, \(243\) can be written as \(\frac{243}{1}\) to satisfy the condition that \(a\) and \(b\) are integers.

Key Concepts

Power of a Power Rule

Understanding the "power of a power rule" is crucial when dealing with expressions like \((3^2)^{5/2}\). This mathematical rule states that when you have an exponent raised to another exponent, you simply multiply the exponents together. For instance, if you have \((a^m)^n\), it becomes \(a^{m \cdot n}\).
When this rule is applied to our problem, \((3^2)^{5/2}\) turns into \(3^{2 \cdot (5/2)} = 3^5\). It's a powerful tool that simplifies handling complex exponential expressions.
The rule helps break down equations into manageable parts, making it easier to calculate with large numbers or fractions. By grasping this concept, you can effectively manage exponential terms and solve problems quickly.

Fractional Exponent

A "fractional exponent" provides a way to express powers and roots together in a single statement, which can simplify equations greatly. When you see an exponent like \(5/2\), the numerator represents the power, and the denominator represents the root. In this expression, \(9^{5/2}\), 5 is the power, and 2 is the root.
This means you first take the square root of 9, then raise the result to the fifth power. Here’s how it works:

• Treat \(9\) as a base being raised to \(5/2\) power.
• Recognize that \(5/2\) means \(\sqrt{9^5}\) or equivalently \((3^2)^{5/2}\).

This process simplifies calculations, especially when dealing with larger powers or more complex roots.
Working with fractional exponents blends the power of two mathematical operations, helping to resolve equations in elegant and efficient ways.

Simplifying Exponents

"Simplifying exponents" is an essential skill for making lengthy or complex expressions more manageable. In solving for \(9^{5/2}\), simplifying is achieved by acknowledging the power of a power rule and fractional exponents.
Follow these steps to simplify smoothly:

• Recognize the base: break it down to a simple form. We converted 9 to \(3^2\).
• Apply the power rules efficiently: use \((3^2)^{5/2}\) to retrieve \(3^5\).
• Calculate the simplified form: compute \(3^5\) which results in 243.

Each of these measures helps in reducing a potentially complicated calculation into straightforward arithmetic
By understanding the components and systematic rules, simplifying exponents becomes a straightforward process that enhances problem-solving skills.