Problem 92 - College Algebra

Question

Simplify using properties of exponents. $$\left(3 x^{\frac{2}{3}}\right)\left(4 x^{\frac{3}{4}}\right)$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\left(3x^{\frac{2}{3}}\right)\left(4x^{\frac{3}{4}}\right)\) is \(12x^{\frac{17}{12}}\).

Step 1: Identify the base and the exponents

First, let's identify the base and the exponents. Here, the base is 'x'. The exponents are \(\frac{2}{3}\) and \(\frac{3}{4}\).

Step 2: Simplify the constants

The constants (3 and 4) multiplying the terms can be simplified independently of the variable 'x'. Thus, we multiply 3 and 4 to get 12.

Step 3: Adding exponents

Next, let's add the exponents of 'x'. According to the rule of exponents, when we are multiplying terms with the same base, we simply add the exponents. So, \(x^{\frac{2}{3}} \times x^{\frac{3}{4}}\) becomes \(x^{\frac{2}{3} + \frac{3}{4}}\)

Step 4: Simplify the exponents

Now we have \(x^{\frac{2}{3} + \frac{3}{4}}\). We first convert the fractions to have the same denominator and then add them. \(\frac{2}{3} + \frac{3}{4}\) becomes \(\frac{8}{12} + \frac{9}{12} = \frac{17}{12}\).

Step 5: Combine the constants and variables

Finally, we combine the constants from step 2 with the simplified variable from step 4 to get the result: \(12x^{\frac{17}{12}}\)