Problem 22 - Algebra and Trigonometry

Question

Evaluate each expression. $$ 3^{-1}-3^{-3} $$

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{8}{27}\).

Step 1: Understand Negative Exponents

Recall that a negative exponent means the reciprocal of the base raised to the corresponding positive exponent. For example, \[a^{-n} = \frac{1}{a^n}\].

Step 2: Evaluate 3^{-1}

Using the rule for negative exponents, \[3^{-1} = \frac{1}{3^1} = \frac{1}{3}\].

Step 3: Evaluate 3^{-3}

Following the same rule for negative exponents, \[3^{-3} = \frac{1}{3^3} = \frac{1}{27}\].

Step 4: Subtract the Values

Now subtract the two fractions obtained earlier: \[\frac{1}{3} - \frac{1}{27}\].To subtract these fractions, find a common denominator. The least common denominator of 3 and 27 is 27.

Step 5: Rewrite the Fractions with a Common Denominator

Rewrite \(\frac{1}{3}\) as \(\frac{9}{27}\), so that both fractions have a denominator of 27: \[\frac{9}{27} - \frac{1}{27}\].

Step 6: Perform the Subtraction

Subtract the numerators: \[\frac{9}{27} - \frac{1}{27} = \frac{8}{27}\].

Step 7: Simplify if Possible

Check if \(\frac{8}{27}\) can be simplified. Since 8 and 27 have no common factors (other than 1), \(\frac{8}{27}\) is already in its simplest form.

Key Concepts

Reciprocal

Negative exponents can be tricky, but they represent a simple concept. When you see a negative exponent, it means you need to find the reciprocal of the base.
The reciprocal of a number is essentially 1 divided by that number. For example, if we have a number like 3, the reciprocal would be \( \frac{1}{3} \). It's just flipping the number to turn it into a fraction.

• For instance, with a negative exponent: \( 3^{-1} \) becomes \( \frac{1}{3^1} \), which simplifies to \( \frac{1}{3} \).
• Similarly, \( 3^{-3} \) translates to \( \frac{1}{3^3} \), making it \( \frac{1}{27} \).

Understanding this is crucial for simplifying expressions with negative exponents, and allows for proper evaluation of expressions like the one given in the exercise.

Fractions

Fractions represent parts of a whole. They consist of a numerator (the top part) and a denominator (the bottom part). In problems involving subtraction of fractions, understanding how they operate helps simplify the process.
For fractions like \( \frac{1}{3} \) and \( \frac{1}{27} \), you need to focus on their numerators and denominators. Here:

• \( \frac{1}{3} \) has a numerator of 1 and a denominator of 3.
• \( \frac{1}{27} \) has a numerator of 1 and a denominator of 27.

To subtract these fractions, the numerators will change while the denominators remain the same once a common denominator is found. Fractions should be managed carefully to avoid mistakes in arithmetic operations.

Common Denominator

Subtraction of fractions requires a common denominator. This is a shared denominator that allows you to subtract the numerators directly.
In our exercise, when we have fractions like \( \frac{1}{3} \) and \( \frac{1}{27} \), they do not share a common denominator by default. The least common denominator (LCD) is the smallest multiple both denominators share. For 3 and 27, the LCD is 27 because 27 is a common multiple for both:

• Convert \( \frac{1}{3} \) to a form with the denominator 27. To do this, multiply both numerator and denominator by 9 to get \( \frac{9}{27} \).

With both fractions now sharing the denominator of 27, you can subtract: \( \frac{9}{27} - \frac{1}{27} = \frac{8}{27} \). Finding common denominators is key to managing and simplifying fractions effectively.