Chapter 4.3, Problem 31WE

To determine

To Simplify: The expression $\left(5x^2\right)\left(2x^3\right)+\left(3x\right)\left(4x^4\right)$

Answer to Problem 31WE

The required answer is $22x^5$ .

Explanation of Solution

Given information:

The expression $\left(5x^2\right)\left(2x^3\right)+\left(3x\right)\left(4x^4\right)$

Concept Used:

Rule of exponents for product of powers states that to multiply two powers with same base, the exponents are added.

For example $a^m\cdot a^n=a^{m+n}$

Commutative property of multiplication states that the product of any numbers remains the same irrespective of the order in which we multiply.

For example:

  $2\cdot 5=5\cdot 2=10$

Associative property of multiplication states that the product of the any numbers remains the same regardless of the grouping of factors.

For example:

  $2\left(5\cdot 3\right)=\left(2\cdot 5\right)\cdot 3=30$

Calculation:

  $\left(5x^2\right)\left(2x^3\right)+\left(3x\right)\left(4x^4\right)$

Commutative and Associative property of multiplication.

  $=\left(5\cdot 2\right)\left(x^2\cdot x^3\right)+\left(3\cdot 4\right)\left(x\cdot x^4\right)$

Using rule of exponents for product of powers

  $\begin{array}{l}=10x^{2+3}+12x^{1+4}\\ =10x^5+12x^5\\ =22x^5\end{array}$