Problem 21 - Algebra and Trigonometry

Question

Evaluate each exponential expression. $$ \frac{2^{3}}{2^{7}} $$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is 1/16.

Step 1: Identify Base and Exponents

The base for both the numerator (2^3) and denominator (2^7) is the same - 2. The exponent in the numerator is 3, and in the denominator is 7.

Step 2: Apply the Quotient Rule of Exponents

According to the quotient rule of exponentials, \(a^{m} / a^{n} = a^{m-n}\). So, we can simplify the expression by subtracting the exponent of the denominator from the exponent of the numerator. Which gives \(2^{3-7} = 2^{-4}\)

Step 3: Convert Negative Exponent to Positive

To convert a negative exponent to a positive exponent, we use the identity \(a^{-n} = 1/a^{n}\). Applying this we get the expression \(2^{-4} = 1/2^{4}\).

Step 4: Evaluate the Expression

Now we just need to evaluate the expression \(1/2^{4}\) which simplifies to 1/16.