Q12. - Algebra 1

Question

Simplify the expression 

${\left(x{y}^4\right)}^6$

Step-by-Step Solution

Answer

The simplified version of ${\left(x{y}^4\right)}^6$ is $x^6{y}^{24}$.

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.

 

That is, ${\left(xy\right)}^m=x^m{y}^m$

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.

 

That is, ${\left(x^m\right)}^n=x^{m\times n}$

Step 3. Simplify the expression.

The given expression is: ${\left(x{y}^4\right)}^6$

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

${\left(x{y}^4\right)}^6=\left(x^6\right){\left(y{}^4\right)}^6$ 

Apply the ‘Law of indices’ for power of index numbers in width="42" height="24" style="max-width: none; vertical-align: -4px;" ${\left(y{}^4\right)}^6$ and simplify further.

$$\begin{gathered} {\left(x{y}^4\right)}^6=\left(x^6\right){\left(y{}^4\right)}^6 \\ =\left(x^6\right)\left(y^{4\times 6}\right) \\ =\left(x^6\right)\left(y^{24}\right) \\ =x^6{y}^{24} \end{gathered}$$