Question

Simplify the expression

${\left(4a^4{b}^{9}c\right)}^2$

Short answer

The simplified version of${\left(4a^4{b}^{9}c\right)}^2$ is $16a^8{b}^{18}{c}^2$.

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:${\left(4a^4{b}^{9}c\right)}^2$

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

${\left(4a^4{b}^{9}c\right)}^2={\left(4\right)}^2{\left(a^4\right)}^2{\left(b^9\right)}^2{\left(c\right)}^2$

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

$$\begin{gathered} {\left(4a^4{b}^{9}c\right)}^2={\left(4\right)}^2{\left(a^4\right)}^2{\left(b^9\right)}^2{\left(c\right)}^2 \\ =16\left(a^{4\times 2}\right)\left(b^{9\times 2}\right)\left(c^2\right) \\ =16\left(a^8\right)\left(b^{18}\right)\left(c^2\right) \\ =16a^8{b}^{18}{c}^2 \end{gathered}$$