Chapter 4.3, Problem 30WE

To determine

To Simplify: The expression $\left(5b^4\right)\left(-3a^{2}b\right)\left(-a^3\right)$

Answer to Problem 30WE

The required answer is $15b^5{a}^5$ .

Explanation of Solution

Given information:

The expression $\left(5b^4\right)\left(-3a^{2}b\right)\left(-a^3\right)$

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:

  $\left(5b^4\right)\left(-3a^{2}b\right)\left(-a^3\right)$

Commutative and Associative property of multiplication.

  $=\left(5\cdot \left(-3\right)\cdot \left(-1\right)\right)\left(b^4\cdot b\right)\left(a^2\cdot a^3\right)$

Using rule of exponents for product of powers

  $\begin{array}{l}=15\left(b^{4+1}\right)\left(a^{2+3}\right)\\ =15b^5{a}^5\end{array}$