Problem 20 - Algebra 1

Question

Evaluate the exponential expression. Write fractions in simplest form $$-6^{0} \cdot \frac{1}{3^{-2}}$$

Step-by-Step Solution

Answer

The evaluated expression is 9.

Step 1: Evaluate the Exponent 0

Evaluate the part \(-6^{0}\). Any nonzero number raised to the zero power equals one, so \(-6^{0}=1\).

Step 2: Evaluate the Negative Exponent

Evaluate the part \(3^{-2}\). The negative exponent means reciprocating the base and make the exponent positive. So, \(3^{-2} = \frac{1}{3^{2}} = \frac{1}{9}\).

Step 3: Perform the Multiplication

Now that the exponential expressions have been evaluated, perform the following multiplication: \(1 \cdot \frac{1}{3^{-2}} = 1 \cdot \frac{1}{\frac{1}{9}} = 1 \cdot 9 = 9.\)

Key Concepts

Exponents

In mathematics, exponents are a way to represent repeated multiplication of a number by itself. Let's say you have a number, known as the base, and you're multiplying it by itself a certain number of times, which is the exponent. For example, in the expression \(2^5\), the number 2 is multiplied by itself five times: \(2 \times 2 \times 2 \times 2 \times 2\). This equals 32. But instead of writing all those multiplies, you use an exponent to simplify. The number being multiplied is called the base, and the exponent is how many times you multiply it by itself. Exponents are not limited to positive integers. They can also be zero or negative, which we'll explore further below.
In our exercise, we evaluated \(-6^0\). Here, remember any nonzero number raised to the zero power is 1. This is a rule in exponents that's crucial to ensuring expressions are simplified correctly.

Negative Exponents

When we talk about negative exponents, they indicate that we need to take the reciprocal of the base raised to the corresponding positive exponent. It's like flipping the base number to the bottom of a fraction and then raising it to the positive version of the negative exponent. For instance, when encountering \(3^{-2}\), you would transform it first to the reciprocal, leading to \(\frac{1}{3^2}\). Then, calculate \(3^2\) which turns into 9, making \(3^{-2}=\frac{1}{9}\).
Negative exponents often lead to a smaller number unless the base is a fraction. Using the original solution steps, once the exponent is positive and in fraction form, calculations become straightforward and manageable.

Evaluating Expressions

Evaluating expressions refers to simplifying or solving them to get a single number or value. To evaluate an exponential expression step-by-step, follow each logical sequence until no further simplification is possible.
For the given exercise, it involves three main steps:

• First, handle each component of the expression separately, starting with immediate exponent evaluations like \(-6^0\), which equates to 1.
• Address negative exponents, converting \(3^{-2}\) to its fraction equivalent, which gives \(\frac{1}{9}\).
• Finally, work through the entire expression by performing the necessary operations, like multiplication. Here it would mean multiplying the evaluated components: \(1 \cdot 9 = 9\).

This process of step-by-step simplification ensures expressions are evaluated consistently and accurately, shedding light on the true essence of the problem posed.