Problem 25 - Algebra 1

Question

Evaluate the exponential expression. Write fractions in simplest form $$-4 \cdot(-4)^{-1}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(-4 \cdot(-4)^{-1}\) is 1.

Step 1: Simplify the exponent

Simplify \((-4)^{-1}\) by recognizing that raising -4 to the power of -1 is the same as taking its reciprocal. The result is \(-1/4\).

Step 2: Multiply by -4

Next, multiply \(-1/4\) by \(-4\). The negative signs cancel each other and we get \(1\).

Key Concepts

Reciprocals

In mathematics, a reciprocal is essentially the flipped version of a fraction or a whole number. When dealing with reciprocals, you're essentially swapping the numerator and the denominator. The reciprocal of any non-zero number 'a' is '1/a'.
For example:

• The reciprocal of 5 is \frac{1}{5}.
• For a fraction \frac{3}{4}, the reciprocal would be \frac{4}{3}.

Reciprocals are often used for dividing fractions, where you multiply by the reciprocal instead.
In our exercise, \((-4)^{-1}\) means we take the reciprocal of -4, which is \-\frac{1}{4}. This step is crucial in simplifying or solving exponential expressions with negative exponents. Remember, any number multiplied by its reciprocal equals 1.

Negative Exponents

Negative exponents can seem intimidating, but they are actually quite simple. A negative exponent signifies that we should take the reciprocal of the base and raise it to the positive of that exponent.

For example, \(a^{-n} = \frac{1}{a^n}\). This means that if you have a number raised to a negative exponent, it can be rewritten as the reciprocal of that number raised to the opposite, positive exponent.

In the context of our exercise, the expression \((-4)^{-1}\) demonstrates this property, transforming it to the reciprocal of -4, or \-\frac{1}{4}. Understanding negative exponents is a key part of working with exponential expressions, making them easier to simplify and evaluate.

Simplification of Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1. During simplification, you'll also check for any cancellation possibilities, especially when multiplying fractions.

In our exercise, after converting the negative exponent expression to a reciprocal, you multiply the two fractions: \-4 \times \-\frac{1}{4}.

It's important to remember that multiplying two negative numbers results in a positive product; hence \(-4 \times -\frac{1}{4} = 1\). The negative signs cancel each other out, simplifying to a clean result of 1, the simplest form of this multiplication problem.