WRITING QUITE COMPLICATED EXPRESSIONS IN THE FORM $\,kx^n\,$
LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
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EXAMPLES:
Question: Write $\,-(3x)(-x)^4\,$ in the form $\,kx^n\,$.
Solution: $\,-3x^5\,$

Why? Keep reading!

Here's the strategy:

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 $\,-2x^6\,$.

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)