Question

Simplify the expression

${\left(-3d^2{f}^{3}g\right)}^2{\left[{\left(-3d^{2}f\right)}^3\right]}^2$

Short answer

The simplified version of${\left(-3d^2{f}^{3}g\right)}^2{\left[{\left(-3d^{2}f\right)}^3\right]}^2$ is $6561d^{16}{f}^{12}{g}^2$.

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. 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:${\left(-3d^2{f}^{3}g\right)}^2{\left[{\left(-3d^{2}f\right)}^3\right]}^2$

Apply the laws of indices and simplify.

$$\begin{gathered} {\left(-3d^2{f}^{3}g\right)}^2{\left[{\left(-3d^{2}f\right)}^3\right]}^2=\left[{\left(-3\right)}^2{\left(d^2\right)}^2{\left(f^3\right)}^2{\left(g\right)}^2\right]{\left[{\left(-3\right)}^3{\left(d^2\right)}^3{\left(f\right)}^3\right]}^2 \\ =\left[9\left(d^{2\times 2}\right)\left(f^{3\times 2}\right)g^2\right]{\left[-27\left(d^{2\times 3}\right)f^3\right]}^2 \\ =\left[9\left(d^4\right)\left(f^6\right)g^2\right]{\left[-27\left(d^6\right)f^3\right]}^2 \\ =\left(9d^4{f}^6{g}^2\right){\left(-27d^6{f}^3\right)}^2 \\ =\left(9d^4{f}^6{g}^2\right)\left[{\left(-27\right)}^2{\left(d^6\right)}^2{\left(f^3\right)}^2\right] \\ =\left(9d^4{f}^6{g}^2\right)\left[729\left(d^{6\times 2}\right)\left(f^{3\times 2}\right)\right] \\ =\left(9d^4{f}^6{g}^2\right)\left[729\left(d^{12}\right)\left(f^6\right)\right] \\ =\left(9d^4{f}^6{g}^2\right)\left(729d^{12}{f}^6\right) \end{gathered}$$

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

$$\begin{gathered} {\left(-3d^2{f}^{3}g\right)}^2{\left[{\left(-3d^{2}f\right)}^3\right]}^2=\left(9d^4{f}^6{g}^2\right)\left(729d^{12}{f}^6\right) \\ =\left(9\times 729\right)\left(d^4\times d^{12}\right)\left(f^6\times f^6\right)\left(g^2\right) \\ =\left(6561\right)\left(d^{4+12}\right)\left(f^{6+6}\right)\left(g^2\right) \\ =6561d^{16}{f}^{12}{g}^2 \end{gathered}$$