Writing Quite Complicated Expressions in the Form kx^n

EXAMPLES:

Question: Write

$\,-(3x)(-x)^4\,$ in the form $\,kx^n\,$.

Solution:

$\,-3x^5\,$

Here's the strategy:

Make three passes through the expression,
figuring out the SIGN, SIZE, and VARIABLE PART.
On the first pass, just figure out the plus/minus sign.
There are five factors of $\,-1\,$ (one outside, four inside);
this is an odd number, so the result is negative.
Here are those five factors:
$\,\overset{\downarrow}{-}(3x)(\overset{\downarrow}{-}x)^{\overset{\downarrow}{4}}\,$
On the second pass, figure out the size of the answer;
you're ignoring all the plus/minus signs, because you took care of them on the first pass.
The size is $\,3\,$:
$\,-(\overset{\downarrow}{3}x)(-x)^4\,$
On the third pass, figure out the power of $\,x\,$.
There are five factors of $\,x\,$, so the variable part is $\,x^5\,$:
$\,-(3\overset{\downarrow}{x})(-\overset{\downarrow}{x})^{\overset{\downarrow}{4}}\,$
Put it all together to get $\,-3x^5\,$.

Question: Write

$\,(-1)^2(-3x)^2(-x)^2\,$ in the form $\,kx^n\,$.

Solution:

$\,9x^4\,$

Sign:
There are six factors of $\,-1\,$;
this is an even number, so the result is positive:
$\,(\overset{\downarrow}{-}1)^{\overset{\downarrow}{2}} (\overset{\downarrow}{-}3)^{\overset{\downarrow}{2}} (\overset{\downarrow}{-}x)^{\overset{\downarrow}{2}} \,$
Size:
The size is $\,9\,$:
$\,(-1)^2(-\overset{\downarrow}{3}x)^{\overset{\downarrow}{2}}(-x)^2\,$
Variable part:
There are four factors of $\,x\,$, so the variable part is $\,x^4\,$:
$\,(-1)^2 (-3\overset{\downarrow}{x})^{\overset{\downarrow}{2}} (-\overset{\downarrow}{x})^{\overset{\downarrow}{2}} \,$
Put it all together to get $\,9x^4\,$.

Question: Write

$\,(-1)^4(-x^3)(-2x)(-x^2)\,$ in the form $\,kx^n\,$.

Solution:

$\,-2x^6\,$

Sign:
There are seven factors of $\,-1\,$;
this is an odd number, so the result is negative:
$\,(\overset{\downarrow}{-}1)^{\overset{\downarrow}{4}} (\overset{\downarrow}{-}x^3) (\overset{\downarrow}{-}2x) (\overset{\downarrow}{-}x^2) \,$
Size:
The size is $\,2\,$:
$\,(-1)^4(-x^3)(-\overset{\downarrow}{2}x)(-x^2)\,$
Variable part:
There are six factors of $\,x\,$, so the variable part is $\,x^6\,$:
$\,(-1)^4 (-\overset{\downarrow}{x}{}^{\overset{\downarrow}{3}}) (-2\overset{\downarrow}{x}) (-\overset{\downarrow}{x}{}^{\overset{\downarrow}{2}})\,$
Put it all together to get $\,-2x^6\,$.

Helpful facts to remember:

$2^5 = 32$ $3^4 = 81$ $3^5 = 243$ $4^3 = 64$ $5^3 = 125$

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Identifying Variable Parts and Coefficients of Terms

Input the exponent using the “ ^ ” key: on my keyboard, it is above the $\,6\,$.
If the answer is (say) $\,3\,$, you must write it as $\,3x^0\,$.
If the answer is (say) $\,3x\,$, you must write it as $\,3x^1\,$.

(an even number, please)