Q17. - Algebra 1

Question

Simplify the expression 

${\left(5x^{2}y\right)}^2{\left(2x{y}^{3}z\right)}^3\left(4xyz\right)$

Step-by-Step Solution

Answer

The simplified version of ${\left(5x^{2}y\right)}^2{\left(2x{y}^{3}z\right)}^3\left(4xyz\right)$ is $800x^8{y}^{12}{z}^4$.

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. State the multiplication rule of Law of indices.

If the two terms have the same base (in this case x) and are to be multiplied together, then their indices are added.

 

In general: $x^m\times x^n=x^{m+n}$

Step 4. Simplify the expression.

The given expression is: ${\left(5x^{2}y\right)}^2{\left(2x{y}^{3}z\right)}^3\left(4xyz\right)$

 

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

${\left(5x^{2}y\right)}^2{\left(2x{y}^{3}z\right)}^3\left(4xyz\right)=\left[{\left(5\right)}^2{\left(x^2\right)}^2{\left(y\right)}^2\right]\left[{\left(2\right)}^3{\left(x\right)}^3{\left(y^3\right)}^3\left(z\right){}^3\right]\left(4xyz\right)$  

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

$$\begin{gathered} {\left(5x^{2}y\right)}^2{\left(2x{y}^{3}z\right)}^3\left(4xyz\right)=\left[{\left(5\right)}^2{\left(x^2\right)}^2{\left(y\right)}^2\right]\left[{\left(2\right)}^3{\left(x\right)}^3{\left(y^3\right)}^3\left(z\right){}^3\right]\left(4xyz\right) \\ =\left[25\left(x^{2\times 2}\right)y^2\right]\left[8x^3\left(y^{3\times 3}\right)z^3\right]\left(4xyz\right) \\ =\left(25x^4{y}^2\right)\left(8x^3{y}^9{z}^3\right)\left(4xyz\right) \end{gathered}$$  

Collect the like terms together and use multiplication rule to simplify further.

$$\begin{gathered} {\left(5x^{2}y\right)}^2{\left(2x{y}^{3}z\right)}^3\left(4xyz\right)=\left(25x^4{y}^2\right)\left(8x^3{y}^9{z}^3\right)\left(4xyz\right) \\ =\left(25\times 8\times 4\right)\left(x^4\times x^3\times x\right)\left(y^2\times y^9\times y\right)\left(z^3\times z\right) \\ =800\left(x^{4+3+1}\right)\left(y^{2+9+1}\right)\left(z^{3+1}\right) \\ =800\left(x^8\right)\left(y^{12}\right)\left(z^4\right) \\ =800x^8{y}^{12}{z}^4 \end{gathered}$$