Problem 27 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$27^{-2} \cdot 27^{-1}$$

Step-by-Step Solution

Answer

\(\frac{1}{19683}\)

Step 1: Apply the Product of Powers Property

When multiplying two powers with the same base, add the exponents. The given expression is \(27^{-2} \times 27^{-1}\). Apply the property: \((-2) + (-1) = -3\). Thus, \(27^{-2} \cdot 27^{-1} = 27^{-3}\).

Step 2: Rewrite with Positive Exponents

To express \(27^{-3}\) with a positive exponent, rewrite it as the reciprocal: \(\frac{1}{27^3}\). This is because \(x^{-n} = \frac{1}{x^n}\).

Step 3: Calculate \(27^3\)

Find \(27^3\) by multiplying: \(27 \times 27 = 729\) and then \(729 \times 27 = 19683\). Therefore, \(27^3 = 19683\).

Step 4: Final Answer with Positive Exponents

Now, substitute back \(27^3\) into the expression with positive exponents: \(\frac{1}{27^3} = \frac{1}{19683}\). This is your final answer.

Key Concepts

Negative Exponents

Negative exponents might seem tricky at first, but they're quite intuitive once you get the hang of them. When a number or variable is raised to a negative exponent, it simply means the reciprocal of that number raised to the corresponding positive exponent. For example, if you see something like \(x^{-n}\), it actually represents \(\frac{1}{x^n}\). This rule simplifies expressions and is essential when trying to convert all exponents to positive ones, a common requirement in algebra.

Let's take a closer look at why this works:

• Mathematically, a negative exponent means dividing, rather than multiplying by the base. So, \(a^{-1}\) is \(\frac{1}{a}\).
• As a general rule, any non-zero base \(b\) raised to the power of zero results in 1, i.e., \(b^0 = 1\), because dividing \(b^1\) by \(b^1\) (same base) inherently cancels out to give 1.
• Continuing from the zero exponent: a negative exponent represents a number one step further into division land from zero: \(a^{-2} = \frac{1}{a^2}\).

Understanding negative exponents helps you transform complex powers into simpler, more manageable fractions, which is especially handy in calculus, physics, and bacterial growth equations.

Product of Powers Property

When you encounter two expressions with the same base being multiplied, you apply the **Product of Powers Property**. This property states that you just add the exponents together. For instance, if you have a base \(a\), raised to an exponent \(m\) and the same base \(a\) raised to \(n\), you multiply the two by simply adding their exponents: \(a^m \times a^n = a^{m+n}\).

Here's why this makes sense:

• Consider \(b^3\) and \(b^2\) — here, both are powers of the same base \(b\).
• Multiplying \(b^3\) by \(b^2\) is \((b \times b \times b) \times (b \times b)\), which equals \(b^5\); that's why their exponents add up: \(3 + 2 = 5\).
• This property is particularly useful in algebra when you're asked to simplify expressions with large powers, saving you from performing unnecessary calculations.

The Product of Powers Property is a key pillar in simplifying and working out equations as it reduces complexity and allows for streamlined calculations.

Positive Exponents

After mastering the concept of negative exponents, transitioning to positive exponents becomes straightforward. Positive exponents denote straightforward multiplication of the base. A positive exponent \(n\) attached to a base \(a\) simply tells you to multiply \(a\) by itself \(n\) times: \(a^n = a \times a \times \, \ldots \, \times a\) (\(n\) times).

A few things to remember about positive exponents:

• They are the most familiar form of exponents you've likely encountered, for example, \(2^3 = 2 \times 2 \times 2 = 8\).
• Calculating with positive exponents remains at the core of basic arithmetic and algebra, feeding into the more complex aspects of math like polynomial equations and scientific notation.
• Understanding positive exponents is crucial when converting negative exponents by taking the reciprocal, as they form the endpoint in these transformations.

Hence, positive exponents not only simplify arithmetic and algebra but also empower you to work backward from more complex negative exponent scenarios.