Chapter 4.3, Problem 19OE

To determine

To Simplify: The expression $\left(-t^3\right){\left(-t\right)}^3$

Answer to Problem 19OE

The required answer is $t^6$ .

Explanation of Solution

Given information:

The expression $\left(-t^3\right){\left(-t\right)}^3$

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:

  $\begin{array}{l}\left(-t^3\right){\left(-t\right)}^3\\ =\left(-t^3\right)\left(-t\right)\left(-t\right)\left(-t\right)\end{array}$

Using rule of exponents for product of powers

  $\begin{array}{l}=-t^3\cdot \left(t^{1+1}\right)\left(-t\right)\\ =-t^3\cdot t^2\cdot \left(-t\right)\\ =-t^{3+2}\cdot \left(-t\right)\\ =-t^5\cdot \left(-t\right)\\ =t^{5+1}\\ =t^6\end{array}$