Problem 75 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. See Example 7. $$ \frac{1}{9^{-5 / 2}} $$

Step-by-Step Solution

Answer

The simplified expression is 243.

Step 1: Apply Negative Exponent Rule

When simplifying expressions with negative exponents, remember the negative exponent rule: \( a^{-n} = \frac{1}{a^{n}} \). Applying this to the expression \( 9^{-5/2} \), we have \( 9^{-5/2} = \frac{1}{9^{5/2}} \). Consequently, the expression becomes \( \frac{1}{\frac{1}{9^{5/2}}} \).

Step 2: Simplify the Fraction

The expression \( \frac{1}{\frac{1}{9^{5/2}}} \) can be simplified by multiplying by the reciprocal. This gives us \( 9^{5/2} \).

Step 3: Convert the Fractional Exponent to Radical Form

Convert the expression \( 9^{5/2} \) into radical form. The fractional exponent \( \frac{5}{2} \) means taking the square root (due to the denominator 2) of 9 and then raising it to the power of 5. This can be rewritten as \( (\sqrt{9})^5 \).

Step 4: Calculate the Root and Power

Find the square root of 9, which is 3. Then raise this result to the power of 5: \((\sqrt{9})^5 = 3^5 \).

Step 5: Compute the Final Result

Calculate \( 3^5 \). Since \( 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 \), the final answer is 243.

Key Concepts

Negative Exponents

Negative exponents may seem complex, but understanding them is quite simple. A negative exponent, like with any exponent, has a base and a power. However, when the exponent is negative, it implies the reciprocal of the base raised to a positive power. For example, the expression \( a^{-n} \) translates to \( \frac{1}{a^n} \). Here’s the twist: this small "negative" piece results in the base being "flipped" into the denominator. Let’s see an example: if we have \( 9^{-5/2} \), we use the rule that \( a^{-n} = \frac{1}{a^n} \). This expression becomes \( \frac{1}{9^{5/2}} \). Negative exponents flip the fraction and make calculations easier sometimes. They provide a way of expressing fractions or simplifying expressions from a certainly different perspective.

Fractional Exponents

Fractional exponents combine powers and roots, merging them into a single notation. Understanding this concept is crucial, especially for simplifying expressions, like our exercise. Fractional exponents may look daunting at first, but they are straightforward when dissected. Consider \( 9^{5/2} \). The fraction’s numerator, 5, signifies the power you will raise the base to, while the denominator, 2, indicates the root you’ll take. It's essentially combining the \( n \)-th root with a power: \( a^{m/n} = \sqrt[n]{a^m} \). That means \( 9^{5/2} \) can be translated into \( (\sqrt{9})^5 \).

• The numerator (5) tells us to raise the base to this power.
• The denominator (2) specifies the square root to take.

This system allows representation of roots and powers compactly and provides flexibility in solving various algebraic problems.

Radical Expressions

Radicals or root expressions are fundamental in math. They provide simplicity when dealing with expressions involving a square root, cube root, or any other form of a radical. The radical symbol \( \sqrt{} \) indicates we're taking a root. Let’s delve deeper:

• \( \sqrt{9} \) denotes the square root of 9, which equals 3, because \( 3 \times 3 = 9 \).
• Note that raising a number to a fraction’s denominator, such as \( \sqrt[n]{a} \), gives you what is called the "n-th root." In our case, the 2 in \( 9^{5/2} \) grounds it as the square root.

The exercise simplifies radicals further by using them in conjunction with powers. For instance, \( (\sqrt{9})^5 \) simplifies as follows: Calculate \( \sqrt{9} = 3 \), then \( 3^5 \) yields 243. Radicals are integral in expressing roots clearly and enable us to transform expressions into different forms for easier manipulation.