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

Want some simpler practice first?   Writing Expressions in the Form $\,kx^n\,$

Ready to move on to more difficult problems?   Writing Quite Complicated Expressions in the Form $\,kx^n\,$

Examples

Question: Write $\,-(3x)^2\,$ in the form $\,kx^n\,.$

Solution:
$-(3x)^2 = (-1)3^2x^2 = -9x^2\,$
or
$\cssId{s14}{\begin{align} -(3x)^2 &= (-1)(3x)(3x)\cr &= (-1)(3\cdot 3)(x\cdot x)\cr &= -9x^2 \end{align}}$

Question: Write $\,-(2x)^3\,$ in the form $\,kx^n\,.$

Solution:
$-(2x)^3 = (-1)2^3x^3 = -8x^3\,$
or
$\cssId{s20}{\begin{align} -(2x)^3 &= (-1)(2x)(2x)(2x)\cr &= (-1)(2\cdot 2\cdot 2)(x\cdot x\cdot x)\cr &= -8x^3 \end{align}}$

Question: Write $\,-(-3x)^2\,$ in the form $\,kx^n\,.$

Solution:
$-(-3x)^2 = (-1)(-3)^2x^2 = -9x^2\,$
or
$\cssId{s26}{\begin{align} -(-3x)^2 &= (-1)(-3x)(-3x)\cr &= (-1)(-3\cdot -3)(x\cdot x)\cr &= -9x^2 \end{align}}$

For mental math, the following thought process can be used:

• How many factors of $-1$ are there? Three (one outside, two inside); this is an odd number, so the answer is negative.
• What's the size of the answer?   $3^2 = 9$
• What's the variable part?   $x^2$
• Put it together to get $\,-9x^2\,.$

Question: Write $\,-(-2x)^3\,$ in the form $\,kx^n\,.$

Solution:
$-(-2x)^3 = (-1)(-2)^3x^3 = 8x^3\,$
or
$\cssId{s41}{\begin{align} -(-2x)^3 &= (-1)(-2x)(-2x)(-2x)\cr &= (-1)(-2\cdot -2\cdot -2)(x\cdot x\cdot x)\cr &= 8x^3 \end{align}}$

For mental math, the following thought process can be used:

• How many factors of $-1$ are there? Four (one outside, three inside); this is an even number, so the answer is positive.
• What's the size of the answer?   $2^3 = 8$
• What's the variable part?   $x^3$
• Put it together to get $\,8x^3\,.$

Helpful facts to remember:

Practice