Problem 22 - Algebra and Trigonometry

Question

Evaluate each exponential expression. $$ \frac{3^{4}}{3^{7}} $$

Step-by-Step Solution

Answer

The value of \(3^{4}/3^{7}\) simplifies to \(1/27\).

Step 1: Identify the Base and Exponents

In our expression, the base is 3 and the exponents are 4 and 7 respectively.

Step 2: Apply the Quotient of Powers Law

Using the quotient of powers law, we can rewrite \(3^{4}/3^{7}\) as \(3^{4-7}\).

Step 3: Simplify the Exponent

Simplify 4-7 to get -3. So, our expression becomes \(3^{-3}\).

Step 4: Interpret Negative Exponent

A negative exponent means to take the reciprocal of the base. Therefore, \(3^{-3}\) is the same as \(1/3^{3}\).

Step 5: Evaluate the Expression

Evaluate \(1/3^{3}\) to get the answer. \(3^{3}\) equals 27, so \(1/3^{3}\) equals 1/27.