Question

Simplify the expression

${\left(-3p^5{t}^6\right)}^4$

Short answer

The simplified version of${\left(-3p^5{t}^6\right)}^4$ is $81p^{20}{t}^{24}$.

Step-by-step solution

Step 1. State the product power law of ‘laws of indices’. 

According to this law, when a product is raised to a power, every factor of the product is raised to the power.

In general:${\left(xy\right)}^m=x^m{y}^m$

Step 2. State the ‘Law of indices’ for power of index numbers.

If a term with a power is itself raised to a power then the powers are multiplied together.

In general:${\left(x^m\right)}^n=x^{m\times n}$

Step 3. Simplify the expression.

The given expression is:${\left(-3p^5{t}^6\right)}^4$

First apply the product power law to simplify the given expression.

${\left(-3p^5{t}^6\right)}^4={\left(-3\right)}^4{\left(p^5\right)}^4{\left(t^6\right)}^4$

Apply the ‘Law of indices’ for power of index numbers and simplify further.

$$\begin{gathered} {\left(-3p^5{t}^6\right)}^4={\left(-3\right)}^4{\left(p^5\right)}^4{\left(t^6\right)}^4 \\ =81\left(p^{5\times 4}\right)\left(t^{6\times 4}\right) \\ =81\left(p^{20}\right)\left(t^{24}\right) \\ =81p^{20}{t}^{24} \end{gathered}$$