Problem 19 - College Algebra

Question

Evaluate each exponential expression. $$ 3^{-3} \cdot 3 $$

Step-by-Step Solution

Answer

The result of evaluating the expression \(3^{-3} \cdot 3\) is \(\frac{1}{9}\).

Step 1: Handle the Negative Exponent

Negative exponent means reciprocal. Change \(3^{-3}\) into its reciprocal expression, which is \(\frac{1}{3^3}\).

Step 2: Simplify the Reciprocal Expression

Now evaluate \(3^3\), which equals 27. So now we have \(\frac{1}{27} \cdot 3\).

Step 3: Perform the Multiplication

Multiplication of \(\frac{1}{27}\) and \(3\) results in \(\frac{3}{27}\).

Step 4: Simplify the Fraction

Reducing \(\frac{3}{27}\) gives \(\frac{1}{9}\).

Key Concepts

Negative Exponents

Negative exponents in mathematics might appear daunting, but they are more intuitive than they seem. When you encounter a negative exponent like in the expression \(3^{-3}\), it signifies a reciprocal. This means that instead of multiplying \(3\) by itself three times, you are flipping the base into a fraction.

To apply this concept, remember:

• An expression with a negative exponent, such as \(a^{-n}\), is equivalent to \(\frac{1}{a^n}\).
• This "flipping" of the expression allows you to work with positive exponents much more easily.

The transformation from \(3^{-3}\) to \(\frac{1}{3^3}\) is a small step that opens the door to solving the entire problem. It is essential to always convert a negative exponent to its reciprocal counterpart for easier calculations.

Reciprocal Expressions

Reciprocal expressions are a vital concept when dealing with negative exponents. A reciprocal essentially flips the base number and changes the exponent to a positive.

For instance, reciprocating \(3^{-3}\) results in \(\frac{1}{3^3}\). This is exactly what we mean by writing it as a reciprocal expression.

Here’s a quick guide to keep in mind:

• The reciprocal of \(a\) is \(\frac{1}{a}\).
• Be sure to apply the reciprocal operation correctly whenever you see a negative exponent.

Once transformed, you can proceed with the arithmetic operations, confident that the expression is now in a more workable form.

Simplifying Fractions

Simplifying fractions is a fundamental process in mathematics, often needed after logical manipulations like performing multiplications or division. When you reach a fraction, such as \(\frac{3}{27}\), your goal should be to make it as simple as possible.

To simplify fractions, you need to find the greatest common divisor (GCD) of the numerator and the denominator. Here's how:

• Divide both the top (numerator) and the bottom (denominator) by their GCD.
• The fraction \(\frac{3}{27}\) simplifies to \(\frac{1}{9}\), as 3 is the greatest common divisor of both numbers.

Simplification rounds off the problem beautifully, making the result neater and easier to understand. Remember, the simpler the fraction, the more elegant your answer is.

Multiplication of Fractions

Multiplying fractions is straightforward once you understand the basics. When faced with multiplying a whole number and a fraction, such as \(\frac{1}{27} \cdot 3\), keep the process smooth and clear.

Here are some handy tips:

• Write the whole number as a fraction. For example, write \(3\) as \(\frac{3}{1}\).
• Multiply the numerators. Multiply the denominators. In our case, it results in \(\frac{1 \cdot 3}{27 \cdot 1} = \frac{3}{27}\).

It is always essential to multiply directly across the fraction lines. Once completed, you can further simplify the result to a cleaner form. Multiplying fractions is generally one of the easier operations in fraction arithmetic, allowing a clear path to the simplified result.