Problem 45 - College Algebra Essentials

Question

Simplify each exponential expression. $$\left(3 x^{4}\right)\left(2 x^{7}\right)$$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is \(6x^{11}\).

Step 1: Identify the base and the exponents

Here, the base is \(x\), while the exponents are 4 and 7. The coefficients are 3 and 2 respectively.

Step 2: Multiply the coefficients

Firstly, multiply the coefficients 3 and 2 together, which equals 6.

Step 3: Apply the law of exponents

According to the law of exponents for multiplication, \(a^n \cdot a^m = a^{n+m}\), where \(a\) is the base and \(n\) and \(m\) are the exponents, apply this rule to the exponents 4 and 7. This yields \(x^{4+7} = x^{11}\).

Step 4: Combine the results

Combine the result from steps 2 and 3 to get the simplified expression, which is \(6x^{11}\).