﻿ Writing Quite Complicated Expressions in the Form kx^n
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 $\,3x^1\,$.

 Write in the form $\,kx^n$ :

 (an even number, please)