Exponent Rules

Exponent Rules

Learning and Practicing Exponent Rules

Rule 1: Combining Like Terms

We are used to simplifying 2x + 3x = 5x, but sometimes it's not that simple.

If you have something like 2x²y + 3x²y = 5x²y it is not always as obvious as to why it's the same.

Basically, if you have the same variables to the same exponent, they are considered like terms and can be added using normal addition rules.

Multiple Choice

Simplify:

9x²y - 10x²y

• -x²y

• x²y

• 19x²y

• These are not like terms.

Multiple Choice

Combine like terms:

4ab³ + 2ab³ - 3ab

• -3ab³

• 3ab³

• 6ab³ - 3ab

• 9ab³

Rule 2: The Product Rule



 $x^a\cdot x^b=x^{\left(a\cdot b\right)}$  

ex.)  $3a^4\cdot2a^6=3\cdot2a^{\left(4+6\right)}=6a^{10}$  

Look out for whole numbers. They just multiply like normal.

Multiple Choice

Simplify:

 $h^2\cdot h^6$  

•  $h^8$  

•  $h^4$  

•  $h^{12}$  

•  $h^{-4}$  

Multiple Choice

Simplify:

 $\left(-2a^2b\right)\cdot\left(7a^3b\right)$  

•  $5a^3b^2$  

•  $-14a^5b^2$  

•  $-14a^5b$  

•  $-14a^6b^2$  

Rule 3: The Power Rule

 $\left(x^a\right)^b=x^{a\cdot b}$  

ex.)  $\left(2a^2\right)^{3\ }=\ 2^3a^{\left(2\cdot3\right)}=8a^6$  

Look out for whole numbers. They work like normal.

 ( $2^3=2\cdot2\cdot2=8$  )

Multiple Choice

Simplify:

 $\left(x^4\right)^5$  

•  $x^{20}$  

•  $x^9$  

•  $x^{12}$  

•  $x^{-1}$  

Multiple Choice

Simplify:

 $\left(-2m^5\right)^2$  

•  $-4m^7$  

•  $-4m^{10}$  

•  $4m^7$  

•  $4m^{10}$  

Rule 4: The Quotient Rule

 $\frac{x^a}{x^b}\ =\ x^{\left(a-b\right)}$  

ex.)  $\frac{6p^5}{2p^2}\ =\ \frac{6}{2}p^{\left(5-2\right)\ }=3p^3$  

Look out for whole numbers. They divide like normal.

Multiple Choice

Simplify:

 $\frac{27x^6}{9x}$  

•  $3x^7$  

•  $3x^5$  

•  $3x^6$  

•  $3x^4$  

Multiple Choice

Simplify:

 $\frac{\left(y^2\right)^3}{y^4}$  

•  $y^2$  

•  $y$  

•  $y^{12}$  

•  $y^{10}$  

Rule 6: The Negative Exponent Rule

 $x^{-a}=\frac{1}{x^a}$  

ex.)  $4z^{-3}=\frac{4}{z^3}$  

Look out for whole numbers. They don't move unless they  have an exponent next to them.

Multiple Choice

Simplify:

 $-5x^{-2}$  

•  $\frac{1}{5x^2}$  

•  $5x^2$  

•  $\frac{-5}{x^2}$  

•  $\frac{1}{-5x^2}$  

Multiple Choice

Simplify:

 $\frac{4k^2}{8k^4}$  

•  $\frac{1}{2k^2}$  

•  $\frac{2}{k^2}$  

•  $\frac{1}{2k^6}$  

•  $\frac{k^{-2}}{2}$  

Rule 7: The Zero Exponent Rule

 $x^0=1$  

ex.)  $8y^0=8\left(1\right)=8$  

Look out for whole numbers. They don't change unless the 0 exponent is directly affecting them.

Multiple Choice

Simplify:

 $7x^0$  

• 1

• 0

• 7

• 7x

Multiple Choice

Simplify:

 $\frac{w^8}{w^8}$  

• 1

•  $w^0$  

•  $w^8$  

•  $w^{16}$