# Writing Expressions in the Form $kx^n$

Note: When you're looking at the math, then:

• $\,(3x)^2\,$ is most easily read aloud as: ‘three ex, (slight pause), squared’
• $\,3x^2\,$ is most easily read aloud as: ‘three, (very slight pause), ex squared’

In the audio for this lesson, some pauses are over-emphasized to help you hear them!

## Examples

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

Solution:
$(3x)^2 = 3^2x^2 = 9x^2\,$
or
\cssId{s10}{\begin{align} (3x)^2 &= (3x)(3x)\cr &= (3\cdot 3)(x\cdot x)\cr &= 9x^2 \end{align}}

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

Solution:
$(2x)^3 = 2^3x^3 = 8x^3\,$
or
\cssId{s16}{\begin{align} (2x)^3 &= (2x)(2x)(2x)\cr &= (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:
$(-3x)^2 = (-3)^2x^2 = 9x^2\,$
or
\cssId{s22}{\begin{align} (-3x)^2 &= (-3x)(-3x)\cr &= (-3\cdot -3)(x\cdot x)\cr &= 9x^2 \end{align}}

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

• It's a negative number to an even power; so, the answer will be positive.
• 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:
$(-2x)^3 = (-2)^3x^3 = -8x^3\,$
or
\cssId{s33}{\begin{align} (-2x)^3 &= (-2x)(-2x)(-2x)\cr &= (-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:

• It's a negative number to an odd power; so, the answer will be negative.
• What's the size of the answer?   $2^3 = 8$
• What's the variable part?   $x^3$
• Put it together to get $\,-8x^3\,.$

$$\cssId{s40}{2^5 = 32}$$ $$\cssId{s41}{3^4 = 81}$$ $$\cssId{s42}{3^5 = 243}$$ $$\cssId{s43}{4^3 = 64}$$ $$\cssId{s44}{5^3 = 125}$$
Input the exponent using the  ‘ ^ ’  key:  on my keyboard, it is above the $\,6\,$.
 Write in the form $\,kx^n\,$: