Problem 42 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ (3 a)^{-4} $$

Step-by-Step Solution

Answer

\(\frac{1}{81a^4}\)

Step 1: Understand the Expression

We have the expression \((3a)^{-4}\). This means we have a base \(3a\) raised to the power of \(-4\). A negative exponent indicates that we will take the reciprocal of the base.

Step 2: Apply the Negative Exponent Rule

The rule for negative exponents is \(x^{-n} = \frac{1}{x^n}\). Apply this to \((3a)^{-4}\), resulting in \(\frac{1}{(3a)^{4}}\).

Step 3: Expand the Expression using Positive Exponents

Now, raise both the 3 and the a to the fourth power separately inside the denominator: \((3a)^4 = 3^4 \cdot a^4\).

Step 4: Calculate Each Power

Calculate \(3^4\) which is \(3 \times 3 \times 3 \times 3 = 81\). Thus, \((3a)^4 = 81a^4\).

Step 5: Write with Positive Exponents

Substitute back into the reciprocal expression: \(\frac{1}{(3a)^{4}} = \frac{1}{81a^4}\). The expression \(\frac{1}{81a^4}\) now has only positive exponents.

Key Concepts

Reciprocal

When you encounter a negative exponent, it's essential to understand that this signals the need to find the reciprocal of the base. A reciprocal, in simple terms, is flipping a number or expression upside down. For example, the reciprocal of a number "x" is "1 over x" or \(\frac{1}{x}\).

In the context of exponents, when you see something like \(x^{-n}\), it means you need to take the reciprocal of \(x\) and raise it to the positive power \(n\). So, \(x^{-n} = \frac{1}{x^n}\).

Applying this to our example, \((3a)^{-4}\), we take the reciprocal of \((3a)\), and we get \(\frac{1}{(3a)^4}\). By doing so, any negative exponent becomes a positive one, making the expression easier to handle in further calculations.

Positive Exponents

Positive exponents tell you how many times to multiply the base by itself. Unlike negative exponents, there's no need to flip the base over. Let's take it step by step with an example from our exercise.

Given \((3a)^{-4}\), after finding its reciprocal, we have \(\frac{1}{(3a)^4}\). Here, \((3a)^4\) implies multiplying \(3a\) by itself four times: \(3a \times 3a \times 3a \times 3a\). To simplify:

• For the number 3 raised to the fourth power: \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).
• For the variable \(a\) raised to the fourth power: \(a^4\).

The expression inside the reciprocal becomes \(81a^4\), emphasizing the neatness positive exponents bring when dealing with powers.

Simplest Form

Reaching the simplest form of an expression ensures that it is both compact and easy to understand. This process often involves simplifying expressions with exponents or fractions.

From our original expression, \((3a)^{-4}\), transforming it with positive exponents and reciprocal gave us the expression \(\frac{1}{81a^4}\).

This version is considered in its simplest form because:

• It contains only positive exponents.
• The numerator and denominator are reduced to their simplest values, meaning there's nothing further to simplify.

Reducing an expression to its simplest form is a valuable skill, especially in algebra, because it makes calculations more manageable and mistakes less likely. Remember, a simple expression is always easier to understand and work with.