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