Problem 10 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ \left(9^{-1}\right)^{2} $$

Step-by-Step Solution

Answer

\(\frac{1}{81}\)

Step 1: Identify the Base and Exponents

For the expression \( (9^{-1})^{2} \), the main base is 9, the first exponent is -1 and the second exponent is 2.

Step 2: Apply the Power of a Power Rule

The power of a power rule states that when raising a power to a power, multiply the exponents. So \( (9^{-1})^{2}=9^{-1*2} = 9^{-2} \).

Step 3: Convert Negative Exponent to Positive

According to the rule of exponents, a negative exponent can be converted into positive by reciprocating the base. So, \( 9^{-2} = \frac{1}{9^{2}} \).

Step 4: Calculate the Expression

Now, \( \frac{1}{9^{2}} \) = \( \frac{1}{81} \).

Key Concepts

Power of a Power Rule

In mathematics, handling exponents can initially seem tricky, but knowing the right rules simplifies everything. One such method is the "Power of a Power Rule." This rule makes it easy to understand and work with expressions that include raising another power. When you see an expression like \((9^{-1})^2\), you are actually looking at two exponents at work.

• The base here is 9.
• There are two exponents: -1 and 2.

To simplify this, according to the Power of a Power Rule, you multiply the exponents together. Multiplying the first exponent \(-1\) with the second exponent \(2\) yields \(-1 \times 2 = -2\). Therefore, \((9^{-1})^{2}\) simplifies to \(9^{-2}\). By applying this rule correctly, we reduce our problem to handling a single exponent, which is much easier to manage.
Understanding this rule lets you streamline expressions quickly without getting lost in complex calculations.

Negative Exponents

Negative exponents might seem strange, but they actually have a very practical purpose. In the world of exponents, a negative exponent is simply a clue to perform a reciprocal.

• A base raised to a negative exponent means that you have to take the reciprocal of the base and then apply the positive exponent.

For example, in the expression \(9^{-2}\), you should think of it as \(\frac{1}{9^2}\). This essentially means that instead of multiplying 9 by itself, you're considering how many times \(9\) fits into 1, based on the exponent.
A good way to remember this idea is: "Flip it to make it positive." Whenever you see a negative exponent, just flip the base and apply the exponent as a positive number. This conversion is crucial for evaluating such expressions accurately.

Evaluating Expressions

Evaluating exponent-based expressions is all about breaking down the components and systematically simplifying them. With the knowledge of the rules, like Power of a Power and negative exponents, you can address these expressions confidently.

• Always start by identifying the base and the exponents involved.
• Use rules to simplify complex parts step-by-step.

Take the expression \(9^{-2}\), as derived earlier. First, convert the negative exponent using the technique of reciprocating. This becomes \(\frac{1}{9^2}\).
Next, perform the arithmetic by evaluating the positive exponent. Here, you calculate \(9^2 = 81\), so the expression \(\frac{1}{9^2}\) becomes \(\frac{1}{81}\). By thoroughly understanding each component and rule, you simplify the process and successfully evaluate the expression without a calculator.