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:  Writing More Complicated Expressions in the Form $\,kx^n\,$

Examples

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 $\,x^6\,$:

  $$(-1)^4 \cssId{s70}{(-\overset{\downarrow}{x}{}^{\overset{\downarrow}{3}})} \cssId{s71}{(-2\overset{\downarrow}{x})} \cssId{s72}{(-\overset{\downarrow}{x}{}^{\overset{\downarrow}{2}})}$$
• Put it all together to get $\,-2x^6\,.$

Helpful facts to remember:

Practice