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
$\cssId{s16}{\begin{align}
(2x)^3 &= (2x)(2x)(2x)\cr
&= (2\cdot 2\cdot 2)(x\cdot x\cdot x)\cr
&= 8x^3
\end{align}}$
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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 $\,-8x^3\,.$
Helpful facts to remember:
$$\cssId{s40}{2^5 = 32}$$ $$\cssId{s41}{3^4 = 81}$$ $$\cssId{s42}{3^5 = 243}$$ $$\cssId{s43}{4^3 = 64}$$ $$\cssId{s44}{5^3 = 125}$$
Practice
Input the exponent using the ‘ ^ ’ key: on my keyboard, it is above the $\,6\,.$