Chapter 4.3, Problem 17E

Solution

To calculate: The error in simplifying the given problem.

The error is that the coefficient is not multiplied by the power. The correct answer for the given expression is $9e^{10x}$.

Given information:

Simplification of an expression,

  $\begin{array}{l}$ {(3e^{5x})}^2=3e^{\left(5x\right)\left(2\right)}$=3e^{10x}\end{array}$.

Calculation: Step 1: Check which property of exponents goes with the given question.

Here, property of power raised to a power meets which states that if a is a base then the power raised to the power of base a gives the product of powers raised to the base a. it means ${\left(a^m\right)}^n=a^{mn}$.

Also, when we have ${\left(x{a}^m\right)}^n=x^n{a}^{mn}$ , where $x$ is any variable.

Step 2: Check what are the coefficients, exponents and base.

• An exponent is a number or letter written above and to the right of a mathematical expression called the base.
• Coefficient is a number multiplied by a variable.
• A base is a number which gets multiplied when using an exponent.

Here, base is $e^{}$ and the coefficient is $3^2=9$.

Calculation to simplify the given expression is,

  $\begin{array}{l}$ {(3e^{5x})}^2=3^2\cdot {(e^{5x})}^2$=9\cdot e^{5x\cdot 2}\\ =9e^{10x}\end{array}$

The correct answer for the given expression is $9e^{10x}$.