﻿ Writing Quite Complicated Expressions in the Form kx^n
WRITING QUITE COMPLICATED EXPRESSIONS IN THE FORM $\,kx^n\,$
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
• PRACTICE (online exercises and printable worksheets)
Want more details, more exercises?
• Want some very basic practice first?   Writing Expressions in the Form $\,kx^n\,$
Here's some medium-difficulty practice:

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:
$\,\cssId{s25}{\overset{\downarrow}{-}}(3x) \cssId{s26}{(\overset{\downarrow}{-}x)^{\overset{\downarrow}{4}}}\,$
• 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 $\,3\,$:
$\,-(\cssId{s30}{\overset{\downarrow}{3}}x)(-x)^4\,$
• On the third pass, figure out the power of $\,x\,$.
There are five factors of $\,x\,$, so the variable part is $\,x^5\,$:
$\,-(3\cssId{s33}{\overset{\downarrow}{x}})\cssId{s34}{(-\overset{\downarrow}{x})^{\overset{\downarrow}{4}}}\,$
• 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:
$\, \cssId{s43}{(\overset{\downarrow}{-}1)^{\overset{\downarrow}{2}}} \cssId{s44}{(\overset{\downarrow}{-}3)^{\overset{\downarrow}{2}}} \cssId{s45}{(\overset{\downarrow}{-}x)^{\overset{\downarrow}{2}}} \,$
• Size:
The size is $\,9\,$:
$\,(-1)^2 \cssId{s48}{(-\overset{\downarrow}{3}x)^{\overset{\downarrow}{2}}}(-x)^2\,$
• Variable part:
There are four factors of $\,x\,$, so the variable part is $\,x^4\,$:
$\,(-1)^2 \cssId{s51}{(-3\overset{\downarrow}{x})^{\overset{\downarrow}{2}}} \cssId{s52}{(-\overset{\downarrow}{x})^{\overset{\downarrow}{2}}} \,$
• 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:
$\, \cssId{s61}{(\overset{\downarrow}{-}1)^{\overset{\downarrow}{4}}} \cssId{s62}{(\overset{\downarrow}{-}x^3)} \cssId{s63}{(\overset{\downarrow}{-}2x)} \cssId{s64}{(\overset{\downarrow}{-}x^2)} \,$
• Size:
The size is $\,2\,$:
$\,(-1)^4(-x^3) \cssId{s67}{(-\overset{\downarrow}{2}x)}(-x^2)\,$
• Variable part:
There are six factors of $\,x\,$, so the variable part is $\,x^6\,$:
$\,(-1)^4 \cssId{s70}{(-\overset{\downarrow}{x}{}^{\overset{\downarrow}{3}})} \cssId{s71}{(-2\overset{\downarrow}{x})} \cssId{s72}{(-\overset{\downarrow}{x}{}^{\overset{\downarrow}{2}})}\,$
• Put it all together to get $\,-2x^6\,$.

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

Master the ideas from this section

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 $\,kx^n$ :