Problem 24 - Algebra 1

Question

Evaluate the exponential expression. Write fractions in simplest form $$8^{-7} \cdot 8^{7}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(8^{-7} \cdot 8^7\) is 1.

Step 1: Apply the property of negative exponents

For \(8^{-7}\), the property of negative exponents can be applied, which gives us \(8^{-7} = \frac{1}{8^7}\). So, our expression becomes \(\frac{1}{8^7} \cdot 8^7\).

Step 2: Simplify the expression

The expression can be simplified as \(\frac{1}{8^7} \cdot 8^7 = \frac{8^7}{8^7}\).

Step 3: Simplify further

Any number divided by itself is equal to 1. So, \(\frac{8^7}{8^7} = 1\)

Key Concepts

Negative Exponents

Understanding negative exponents is essential for simplifying exponential expressions. A negative exponent indicates that the number is not really multiplying, it's dividing. When you have an expression like \(8^{-7}\), it tells you to take the reciprocal of the base number to the positive power of the exponent. In our exercise, \(8^{-7}\) becomes \(\frac{1}{8^7}\). This means that instead of multiplying by 8 seven times, you are dividing 1 by 8 seven times.

One way to remember how negative exponents work is to think of them as instructions for where the base number should go—in the numerator (top) or denominator (bottom) of a fraction. A negative exponent moves the base from where it currently is to the other side of the fraction line. If it's on top (in the numerator), it moves to the bottom (in the denominator), and vice versa, making it a positive exponent.

Exponential Equations

Exponential equations often require you to manipulate the terms to simplify or solve for a variable. In the step by step solution we've examined, the equation \(8^{-7} \cdot 8^{7}\) is an example of an exponential equation. To simplify such equations, we utilize the property that when you multiply expressions with the same base, you add their exponents. However, in this case, our exponents are additive inverses; one is negative and the other positive.

What does this mean? When you have the same base with a positive exponent and a negative exponent of equal magnitude, like \(8^{7}\) and \(8^{-7}\), they essentially cancel each other out. In our equation, this creates a fraction with identical numerator and denominator, leading to the final simplified solution of just 1.

Simplifying Fractions

Simplifying fractions is a vital skill when working with numeric or algebraic expressions. To simplify a fraction, you need to find the greatest common factor (GCF) of both the numerator and the denominator and divide both by this number. But if the numerator and denominator are the same, as in the fraction \(\frac{8^7}{8^7}\) from our exercise, the GCF is the entire numerator or denominator. When any nonzero number is divided by itself, the result is always 1.

So, simplifying fractions can sometimes be very straight-forward, as it is in the case of fractions containing exponential expressions with the same bases and exponents. This is an example of a more general principle, where any quantity, not just numbers, divided by itself equals 1, provided it is not zero. This concept is fundamental in mathematics and is frequently used to simplify and solve equations.