Problem 91 - College Algebra

Question

Simplify using properties of exponents. $$\left(7 x^{\frac{1}{3}}\right)\left(2 x^{\frac{1}{4}}\right)$$

Step-by-Step Solution

Answer

The simplified form of \(\left(7 x^{\frac{1}{3}}\right)\left(2 x^{\frac{1}{4}}\right)\) is \(14x^{\frac{7}{12}}\).

Step 1: Multiply coefficients

Multiply the coefficients. We have 7 and 2 as coefficients. Multiply them and write down the result. So, \(7 \times 2 = 14.\)

Step 2: Use properties of exponent

Apply the rule for when the bases are the same and we are multiplying: we add the exponents. So, \(x^{\frac{1}{3}} \times x^{\frac{1}{4}}\) becomes \(x^{\frac{1}{3} + \frac{1}{4}}.\)

Step 3: Simplifying the exponent

Simplify the exponent by calculating the sum \(\frac{1}{3} + \frac{1}{4}\). Find a common denominator of 3 and 4, which is 12, and do the addition. The simplification results in \(\frac{7}{12}\).

Step 4: Write the final simplified expression

Combine the results of step 1 and step 3 to obtain the final simplified expression. The coefficient 14 (from step 1) is multiplied with the base x raised to the power \(\frac{7}{12}\) (from step 3). Our final result is \(14x^{\frac{7}{12}}\).