Problem 11 - Precalculus Functions and Graphs

Question

Determine the number that can be used in place of the question mark to make the equation true. $$3^{?}=\frac{1}{81}$$

Step-by-Step Solution

Answer

The number is -4.

Step 1: Understand the equation

Given the equation \[ 3^{?} = \frac{1}{81} \] it is required to find the exponent that makes this equation true.

Step 2: Express the fraction as a power of 3

Recognize that 81 can be expressed as a power of 3: \[ 81 = 3^4 \] Thus, the equation becomes \[ 3^{?} = \frac{1}{3^4} \]

Step 3: Use properties of exponents

Recall that \( \frac{1}{3^4} \) can be rewritten using negative exponents: \[ \frac{1}{3^4} = 3^{-4} \]

Step 4: Match the exponents

Now, the equation is \[ 3^{?} = 3^{-4} \] To make the bases and exponents equal, the exponent must be -4.

Key Concepts

Negative Exponents

When dealing with exponents, you might come across negative exponents. Understanding negative exponents is essential as they show up often in math problems.
A negative exponent indicates that the base is on the wrong side of a fraction. To correct this, you move the base to the other side and make the exponent positive.
For example, \(\frac{1}{a^n} = a^{-n}\).
This means if you see a fraction, you can rewrite it with a negative exponent, making calculations easier. Always remember, negative exponents do not mean that the number itself is negative, just that it is reciprocated.

Expressing Fractions as Powers

Fractions can often be written as powers or exponents, which simplifies many calculations.
To express a fraction as a power, recognize the base and the exponent. For instance, in our exercise, \(\frac{1}{81}\) can be written as \(\frac{1}{3^4}\).
By acknowledging that \(\frac{1}{3^4}\) is equivalent to \(\frac{1}{81}\), we are able to simplify the equation to a form that is easier to solve.
This method is particularly useful when solving exponential equations, as it allows us to work with consistent bases and apply exponent rules more effectively.

Properties of Exponents

Several key properties of exponents are crucial to solving exponential equations. Let’s look at a few properties used in the given exercise:

• Product of Powers: \(\text{If } a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(\text{If } \frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \(\text{If } (a^m)^n = a^{m \times n}\)
• Negative Exponents: \(\text{If } a^{-n} = \frac{1}{a^n}\)

These properties allow us to transform equations into more workable forms. In our exercise, the property \(\frac{1}{a^n} = a^{-n}\) was particularly helpful in rewriting the fraction \(\frac{1}{81}\) as \(\frac{1}{3^4} = 3^{-4}\). Understanding and applying these properties makes solving exponent equations a breeze.

Solving for Exponents

Solving for exponents may seem challenging, but it becomes much simpler if broken down into steps.
Here’s a brief summary of how to solve such problems using our example: \(\text{Step 1: Given the equation } 3^x = \frac{1}{81}\).
\(\text{Step 2: Express 81 as a power of 3 } (81 = 3^4)\).
\(\text{Step 3: Rewrite the equation as } 3^x = \frac{1}{3^4}\).
\(\text{Step 4: Use the property of negative exponents to transform the fraction: } \frac{1}{3^4} = 3^{-4}\).
\(\text{Step 5: With the equation now as } 3^x = 3^{-4}, \text{it is clear that } x = -4\).
By consistently applying logical steps and exponent properties, solving for exponents becomes much more manageable. Take your time to understand these steps, and it will soon feel effortless.