Problem 18 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ \frac{3}{3^{-2}} $$

Step-by-Step Solution

Answer

The expression \(\frac{3}{3^{-2}}\) simplifies to \(27\).

Step 1: Understanding Negative Exponents

First, we need to understand that a negative exponent means we take the reciprocal of the base and change the exponent to positive. Therefore, for any number with a negative exponent like \(a^{-n}\), it can be rewritten as \(\frac{1}{a^n}\).

Step 2: Rewrite the Denominator

In the expression \(\frac{3}{3^{-2}}\), let's focus on the denominator \(3^{-2}\). By applying the rule for negative exponents, we rewrite it as \(\frac{1}{3^2}\). This transforms the original expression to \(\frac{3}{\frac{1}{3^2}}\).

Step 3: Simplify the Complex Fraction

A complex fraction can be simplified by multiplying by the reciprocal of the denominator. In this case, multiply \(3\) by \(3^2\), so it becomes \(3 \times 3^2\).

Step 4: Calculate the Product

Now, calculate the product of \(3 \times 3^2\). \(3^2\) is equal to \(9\), so the expression becomes \(3 \times 9\).

Step 5: Final Calculation

Multiply \(3\) by \(9\) to get \(27\). Thus, the simplified result of the expression \(\frac{3}{3^{-2}}\) is \(27\).

Key Concepts

Understanding Complex Fractions

A complex fraction is a fraction where the numerator, the denominator, or both have fractions within them. They might seem intimidating at first, but simplifying them makes solving these expressions straightforward.
Consider the expression \( \frac{3}{3^{-2}} \), which we had earlier. This is a simple instance of complex fractions. The numerator is just \(3\) while the denominator is another fraction if we apply the rules for negative exponents.
To handle complex fractions, the essential step is to simplify by combining the layers of fractions. This involves converting the fraction into a simpler form so it can be easily calculated. Typically, this approach revolves around multiplying by the reciprocal of the denominator, which eliminates the fraction in the denominator.
As we learned, once you rewrite the negative exponent, you get a more straightforward fraction, which leads us to understand how to simplify the expression further.

Defining and Using Reciprocals

The concept of reciprocals is quite handy, especially when dealing with fractions. In mathematics, the reciprocal of a number \(a\) is simply \(\frac{1}{a}\).
The reciprocal turns division into multiplication, which is simpler to compute. For example, if you have a fraction like \( \frac{1}{3^2} \), its reciprocal is simply \(3^2\) or \(9\) when calculated.
In the context of our expression \( \frac{3}{3^{-2}} \), the reciprocal helps simplify the complex fraction.
By taking the reciprocal of \(3^{-2}\), it becomes \(\frac{1}{3^2}\), and when we multiply \(3\) by this reciprocal, we eliminate the fractions in the denominator effortlessly. Reciprocals are vital for simplifying complex fractions efficiently.

Simplifying Expressions Involving Exponents

Simplifying expressions is an important skill in algebra, particularly with exponents. When faced with a negative exponent, such as \(3^{-2}\), you start by converting it. Negative exponents can make things seem more complicated than they are.
To simplify an expression with a negative exponent, use the reciprocal as previously discussed. Convert \(3^{-2}\) to \(\frac{1}{3^2}\). This allows you to clear the negative exponent and form a fraction.
Once rewritten, you can simplify further by performing multiplication across the fractions. Thus, \( \frac{3}{\frac{1}{3^2}} \) simplifies to \(3 \times 3^2\). Calculate \(3^2 = 9\), and finally compute \(3 \times 9 = 27\).
Simplifying expressions with exponents, especially negatives, becomes much clearer with practice. Always remember to apply these basic rules: convert to simpler fractions and multiply to eliminate complex fractions.