# Writing Quite Complicated Expressions in the Form $kx^n$

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

$$\cssId{s75}{2^5 = 32}$$ $$\cssId{s76}{3^4 = 81}$$ $$\cssId{s77}{3^5 = 243}$$ $$\cssId{s78}{4^3 = 64}$$ $$\cssId{s79}{5^3 = 125}$$
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 $\,kx^n\,$: