# Writing More Complicated Expressions in the Form $\,kx^n$

Want some simpler practice first? Writing Expressions in the Form $\,kx^n\,$

Ready to move on to more difficult problems? Writing Quite Complicated Expressions in the Form $\,kx^n\,$

## Examples

Question: Write $\,-(3x)^2\,$ in the form $\,kx^n\,.$

Solution:

$-(3x)^2 = (-1)3^2x^2 = -9x^2\,$

or

$\cssId{s14}{\begin{align}
-(3x)^2 &= (-1)(3x)(3x)\cr
&= (-1)(3\cdot 3)(x\cdot x)\cr
&= -9x^2
\end{align}}$

Question: Write $\,-(2x)^3\,$ in the form $\,kx^n\,.$

Solution:

$-(2x)^3 = (-1)2^3x^3 = -8x^3\,$

or

$\cssId{s20}{\begin{align}
-(2x)^3 &= (-1)(2x)(2x)(2x)\cr
&= (-1)(2\cdot 2\cdot 2)(x\cdot x\cdot x)\cr
&= -8x^3
\end{align}}$

Question: Write $\,-(-3x)^2\,$ in the form $\,kx^n\,.$

Solution:

$-(-3x)^2 = (-1)(-3)^2x^2 = -9x^2\,$

or

$\cssId{s26}{\begin{align}
-(-3x)^2 &= (-1)(-3x)(-3x)\cr
&= (-1)(-3\cdot -3)(x\cdot x)\cr
&= -9x^2
\end{align}}$

For mental math, the following thought process can be used:

- How many factors of $-1$ are there? Three (one outside, two inside); this is an odd number, so the answer is negative.
- What's the size of the answer? $3^2 = 9$
- What's the variable part? $x^2$
- Put it together to get $\,-9x^2\,.$

Question: Write $\,-(-2x)^3\,$ in the form $\,kx^n\,.$

Solution:

$-(-2x)^3 = (-1)(-2)^3x^3 = 8x^3\,$

or

$\cssId{s41}{\begin{align}
-(-2x)^3 &= (-1)(-2x)(-2x)(-2x)\cr
&= (-1)(-2\cdot -2\cdot -2)(x\cdot x\cdot x)\cr
&= 8x^3
\end{align}}$

For mental math, the following thought process can be used:

- How many factors of $-1$ are there? Four (one outside, three inside); this is an even number, so the answer is positive.
- What's the size of the answer? $2^3 = 8$
- What's the variable part? $x^3$
- Put it together to get $\,8x^3\,.$

Helpful facts to remember:

$$\cssId{s51}{2^5 = 32}$$ $$\cssId{s52}{3^4 = 81}$$ $$\cssId{s53}{3^5 = 243}$$ $$\cssId{s54}{4^3 = 64}$$ $$\cssId{s55}{5^3 = 125}$$## Practice

Input the exponent using the ‘ **^** ’ key: on my keyboard, it is above the $\,6\,.$