Question:
Write
$\,-(3x)(-x)^4\,$ in the form $\,kx^n\,$.
Solution:$\,-3x^5\,$
Why?Keep reading!
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 $\,x^6\,$:
$\,(-1)^4
\cssId{s70}{(-\overset{\downarrow}{x}{}^{\overset{\downarrow}{3}})}
\cssId{s71}{(-2\overset{\downarrow}{x})}
\cssId{s72}{(-\overset{\downarrow}{x}{}^{\overset{\downarrow}{2}})}\,$
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 $\,3x^1\,$.