Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(\frac{15(x+3)^{3}}{9(x+3)^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{5}{3}(x+3)\).

Step 1: Rewrite the expression

The expression given is \(\frac{15(x+3)^{3}}{9(x+3)^{2}}\). We can start by expressing each term in the simplest form possible, which in this case involves breaking down the 15/9 fraction into its simplest form, and rewriting the terms involving the x+3 base.

Step 2: Simplify the Numerical Fraction

15 and 9 are both divisible by 3. If we divide both the numerator and the denominator by 3, then we simplify the fraction from 15/9 to 5/3.

Step 3: Apply the Rules of Exponents

Next, we look at \((x+3)^{3}\) and \((x+3)^{2}\). In any scenario that involves dividing identical bases with different exponents, we subtract the exponent of the denominator from the exponent of the numerator. Therefore, \(\frac{(x+3)^{3}}{(x+3)^{2}} = (x+3)^{3-2} = (x+3)\).

Step 4: Combine all simplified terms

Now we can combine the simplified terms from Step 2 and Step 3. This gives us the simplified expression: \(\frac{5}{3}(x+3)\).