Problem 85 - Algebra for College Students

Question

Is the following simplification process correct? $$ \left(3^{-2}\right)^{-1}=\left(\frac{1}{3^{2}}\right)^{-1}=\left(\frac{1}{9}\right)^{-1}=\frac{1}{\left(\frac{1}{9}\right)^{1}}=9 $$ Could you suggest a better way to do the problem?

Step-by-Step Solution

Answer

The simplification is correct, but using the power of a power rule is more efficient.

Step 1: Examine the Original Expression

The original expression is \((3^{-2})^{-1}\). We need to simplify this expression step by step without making any mistakes.

Step 2: Apply the Power of a Power Rule

According to the power of a power rule, \((a^m)^n = a^{m\cdot n}\). Applying this to \((3^{-2})^{-1}\), we get \(3^{-2 \cdot (-1)} = 3^2\).

Step 3: Simplify the Expression

We calculated \(3^{(-2) \times (-1)}\) as \(3^2\). Now we know that \(3^2 = 9\). Thus the simplified expression results in 9.

Step 4: Verify the Simplification

The calculation based on the rule was directly \(3^2 = 9\), which is simpler and avoids unnecessary steps like inverting fractions. Both processes result in the same correct answer, but applying the power of a power rule is more straightforward.

Key Concepts

Power of a Power Rule

When dealing with exponentiation, one vital concept to understand is the power of a power rule. This rule helps us simplify expressions where exponents are raised to another exponent. According to the power of a power rule, \[(a^m)^n = a^{m \cdot n}\]This means that when you have an exponent raised to another exponent, you can multiply the exponents to simplify the expression. For example, if you start with an expression like \((3^{-2})^{-1}\), using this rule, you multiply the exponents: - \((-2) \times (-1)\) results in \(2\).Thus, the expression simplifies down to \(3^2\). This approach is efficient and helps avoid additional steps such as inverting fractions. Remember, simplifying with the power of a power rule is both quicker and reduces potential errors in calculation.

Simplifying Expressions

Simplifying mathematical expressions makes them more manageable and understandable. It often involves using rules of algebra to condense expressions into their simplest form. When faced with an expression that seems complicated, like \((3^{-2})^{-1}\), starting by applying established rules such as the power of a power rule can lead to a simpler form.After applying the power of a power rule, you've reduced the exponents multiplication: - \(3^2\) becomes the simplified form. - Evaluate this further: \(3^2\) calculates to \(9\).This approach of focusing on simplification helps in solving problems faster, reduces errors, and improves understanding. Always look for components of the expression that can be consolidated, canceled, or multiplied using fundamental algebraic rules. This is the essence of simplifying expressions, allowing for clear and concise solutions.

Negative Exponents

Negative exponents can often appear intimidating, but are quite simple once you understand their purpose. A negative exponent indicates that the base should be reciprocated in order to convert it to a positive exponent. For example, \(3^{-2}\) translates to:- \(\frac{1}{3^2}\).This rule is useful when simplifying expressions where negative exponents are present. By recognizing a negative exponent, you can make the expression positive and easier to work with.Returning to our example, if you initially work through the expression \((3^{-2})^{-1}\):- Convert to positive exponents if needed, but applying the power of a power rule generally simplifies to positive results directly, like yielding \(3^2 = 9\).In summary, never fear negative exponents. Their use helps in transforming complex expressions into more straightforward, positive ones. This clarity allows you to continue with further mathematical operations smoothly.