Problem 89 - College Algebra Essentials

Question

Perform the indicated operation or operations. $$\frac{(2 x-7)^{5}}{(2 x-7)^{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \((2x-7)^{2}\).

Step 1: Recognize base and powers

In the given expression \(\frac{(2 x-7)^{5}}{(2 x-7)^{3}}\), the base is (2x-7) which is being raised to the power of 5 in the numerator and 3 in the denominator.

Step 2: Apply rule of exponents

The rule of exponents states that when you're dividing expressions with the same base, you subtract the exponent in the denominator from the exponent in the numerator, i.e., \(a^{m}/a^{n} = a^{m-n}\). Here, the base 'a' is (2x-7), 'm' is 5, and 'n' is 3. Thus, \(\frac{(2 x-7)^{5}}{(2 x-7)^{3}} = (2x-7)^{5-3}\)

Step 3: Compute final expression

After subtracting the exponents, the final expression is \((2x-7)^{2}\)