Problem 46 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ 9^{-1} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression $$9^{-1}$$ using only positive exponents. Answer: $$\frac{1}{9}$$

Step 1: Understand the negative exponent rule

The rule for negative exponents states that: $$a^{-n} = \frac{1}{a^n}$$ where 'a' is the base and 'n' is the exponent. When a number has a negative exponent, it can be rewritten as a fraction with the base raised to the corresponding positive exponent in the denominator.

Step 2: Apply the negative exponent rule to the given expression

Now, let's apply the rule for negative exponents to the given expression: $$9^{-1}$$ Using the rule, we can rewrite this expression as: $$\frac{1}{9^1}$$

Step 3: Simplify the expression

As a final step, let's simplify the expression: $$\frac{1}{9^1} = \frac{1}{9}$$ Therefore, the expression $$9^{-1}$$ can be rewritten using only positive exponents as: $$\frac{1}{9}$$

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They don't require an equal sign, which makes them different from equations. When variables are included, expressions can represent unknown quantities.
In the world of algebra, understanding expressions is crucial as they are the foundation upon which equations and inequalities are built. For example, the expression \(9^{-1}\) is an algebraic expression that includes a negative exponent on the number 9. Understanding how to manipulate these expressions by applying mathematical rules, such as exponent rules, helps in solving them or altering them for clarity or computation.

Fraction Simplification

Fraction simplification is all about making a fraction as simple as possible. This typically means presenting it in its lowest terms by ensuring the numerator and denominator have no common factors other than 1.

In the context of expressions with exponents, simplification can involve converting expressions with negative exponents into fractions. For instance, the expression \(9^{-1}\) becomes \(\frac{1}{9}\) once the negative exponent rule is applied. This is already in its simplest form since there are no common factors between the numerator and the denominator.

• Ensures clarity in mathematical operations
• Makes the expression easier to work with in further calculations

Always consider whether the fraction can be reduced further, but in this case, \(\frac{1}{9}\) is as simple as it gets.

Exponent Rules

Exponent rules are crucial for manipulating expressions involving powers. They help in rewriting and simplifying expressions to make calculations straightforward.
When you encounter a negative exponent like in \(9^{-1}\), the rule is to rewrite it as a fraction: \(a^{-n} = \frac{1}{a^n}\). Here, the base remains the same, while the exponent becomes positive in the denominator. This ability to transform negative exponents into positive ones simplifies computation and brings clarity.
Here's a quick summary of some important exponent rules to remember:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{mn}\)
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)
• Zero Exponent: \(a^0 = 1\) \((a eq 0)\)

These rules are foundational in algebra, helping to reshape and calculate expressions with ease.