Problem 76 - 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: Understand the Negative Exponent Rule

Negative exponents indicate that the base should be on the opposite side of the fraction line. In other words, if you have \(n^{-a}\), it can be rewritten as \(\frac{1}{n^a}\). Similarly, \(\frac{1}{n^{-a}}\) becomes \(n^a\).

Step 2: Apply the Negative Exponent Rule

Given the expression \(\frac{1}{9^{-5/2}}\), apply the negative exponent rule: \(9^{-5/2}\) moves to the numerator and becomes \(9^{5/2}\). Thus, \(\frac{1}{9^{-5/2}} = 9^{5/2}\).

Step 3: Simplify the Expression Using the Power of a Power Rule

The expression \(9^{5/2}\) can be simplified further. Recognize that 9 can be written as \(3^2\), so \(9^{5/2} = (3^2)^{5/2}\). Use the power of a power rule \((a^m)^n = a^{mn}\): \[(3^2)^{5/2} = 3^{2 \times (5/2)} = 3^5\].

Step 4: Calculate the Final Expression

Compute \(3^5\). This equals \(3 \times 3 \times 3 \times 3 \times 3 = 243\).

Key Concepts

Exponent Rules

Exponent rules are a set of guidelines that help us manipulate expressions involving powers or indices. Understanding and applying these rules correctly can make working with polynomials and algebraic expressions much simpler.

• Negative Exponent Rule: If an exponent is negative, it indicates that we take the reciprocal of the base. For example, if you see \(n^{-a}\), this is the same as \(\frac{1}{n^a}\). To flip the scenario, \(\frac{1}{n^{-a}}\) equals \(n^a\).
• Power of a Power Rule: When you have an expression like \((a^m)^n\), this equals \(a^{mn}\). This rule is particularly useful in simplifying expressions with compound exponents.
• Product and Quotient of Powers: This rule states that \(a^m \times a^n = a^{m+n}\) and \(\frac{a^m}{a^n} = a^{m-n}\).

These rules apply to any type of base – numbers, variables, or even complex terms.
Make sure to understand them well as they are the core foundation of manipulating expressions.

Simplifying Expressions

Simplifying expressions means condensing them into a neater form without changing their value. It's a crucial skill in algebra and involves a step-by-step approach.
Begin by identifying patterns or operations such as multiplication, division, and utilizing exponent rules. For example, with the expression \(\frac{1}{9^{-5/2}}\), the negative exponent rule was applied to move \(9^{-5/2}\) from the denominator to the numerator as \(9^{5/2}\). This step simplified the problem significantly.

• Start by applying any applicable exponent rules.
• Next, look for opportunities to combine like terms, if applicable.
• Convert complicated looking bases into simpler forms if possible.

Using these techniques systematically ensures that expressions are reduced to their simplest form more easily and accurately. A thorough understanding of how to simplify expressions will make solving equations and other algebraic tasks much more manageable.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations such as addition, subtraction, multiplication, division, and exponents.
Algebraic expressions are used to represent relationships and changes between variables. For example, in the expression \(9^{5/2}\), "9" is a base, which can represent a number or variable, and "5/2" is the exponent that shows how many times the base is used in a multiplication.
Working with algebraic expressions often involves:

• Recognizing patterns such as distributive, associative, and commutative properties.
• Applying the correct order of operations, also known as PEMDAS (parentheses, exponents, multiplication and division, and addition and subtraction).
• Simplifying by combining like terms and using factorization when possible.

Understanding how to handle these expressions is fundamental in solving equations and understanding more complex mathematical concepts. By practicing these principles, you are better prepared to tackle a variety of algebraic challenges.