Problem 19 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 9^{-\frac{1}{2}} $$

Step-by-Step Solution

Answer

\(9^{-\frac{1}{2}} = \frac{1}{3}\).

Step 1: Understand the Task

We need to express the power \( 9^{-rac{1}{2}} \) as a rational number in its simplest form. This involves calculating the power and simplifying the result.

Step 2: Rewrite the Negative Exponent

Recall that a negative exponent indicates the reciprocal of the base with the opposite exponent. Thus, \( a^{-n} = \frac{1}{a^n} \). For \( 9^{-rac{1}{2}} \), we write it as \( \frac{1}{9^{\frac{1}{2}}} \).

Step 3: Simplify the Expression

The next step is to simplify \( 9^{\frac{1}{2}} \). The fraction exponent \( \frac{1}{2} \) corresponds to the square root. Thus, \( 9^{\frac{1}{2}} = \sqrt{9} \). Since the square root of 9 is 3, we have \( \sqrt{9} = 3 \).

Step 4: Substitute Back to Simplify

Substitute the value back into the expression: \( \frac{1}{9^{\frac{1}{2}}} = \frac{1}{3} \).

Step 5: Simplify the Rational Form

The expression \( \frac{1}{3} \) is already in its simplest rational form.

Key Concepts

Negative Exponents

Negative exponents can be confusing at first, but they simply express the idea of taking the reciprocal of a number raised to a positive exponent. Let's break it down:

• When you see a negative exponent, like in the expression \( a^{-n} \), it tells you to take the reciprocal of \( a \) raised to the positive exponent \( n \). So, \( a^{-n} = \frac{1}{a^n} \).
• This rule helps you transform complex expressions into simpler ones. By writing the negative exponent as a fraction, you flip the base number, making it more understandable and easier to work with. In this case, \( 9^{-rac{1}{2}} \) becomes \( \frac{1}{9^{\frac{1}{2}}} \).

Whenever you encounter a negative exponent, remember that it's just a signal to find a reciprocal. This simple shift in perspective can significantly simplify your calculations and lead you to the correct solution.

Fractional Exponents

Fractional exponents are another way to express roots, which might make them seem complex. But they're actually quite straightforward once you understand the relationship between exponents and roots.

• The fraction in the exponent refers to the root of the base. For example, \( a^{\frac{1}{n}} \) represents the \( n \)-th root of \( a \). In the exercise, \( 9^{\frac{1}{2}} \) is equivalent to the square root of \( 9 \), or \( \sqrt{9} \).
• Breaking down the expression helps identify the operations needed and makes it less intimidating. For instance, understanding that the fraction \( \frac{1}{2} \) indicates a square root facilitates solving the problem. Knowing this relationship helps you convert between exponential and root forms quickly.

Thus, converting fractional exponents into root expressions is a key step in evaluating and simplifying these types of problems. Once you see this link, it makes problems involving roots much more approachable.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra and helps make complex problems more manageable. Let's discuss how you can simplify expressions involving exponents.

• When given an expression like \( 9^{-\frac{1}{2}} \), your goal is to express it in its simplest form. By converting negative and fractional exponents step by step, you clarify the calculations needed.
• Start by changing the negative exponent into a reciprocal, then evaluate the fractional exponent as a root, such as finding \( \sqrt{9} = 3 \). Then, substitute back to get the final simplified rational expression \( \frac{1}{3} \).

By breaking down each step and simplifying incrementally, you simplify not just the expression but also the process. This approach clarifies the path to the solution, ensuring that you arrive at the simplest rational form with confidence.