Rules of Exponentiation

The rules of exponentiation are rules that help us perform operations like addition, subtraction, multiplication, and division with powers. In certain exercises, if the rules of exponentiation are not used correctly, it will be very difficult to find the solution, so it's important to know them.

Don't worry! These aren't complicated rules. If you make an effort to understand them and practice enough, you'll be able to apply them easily.

$a^n\cdot a^m=a^{n+m}$

If we multiply powers with the same base, the exponent of the result will be equal to the sum of the exponents.

$5^2\cdot5^3=5^{2+3}=5^5$

$7^{X+1}\cdot7^{2X+2}=7^{3X+3}$

$X^4\cdot X^5=X^{4+5}=X^9$

$\frac{a^n}{a^m}=a^{n-m}$

$a≠0$

When we divide powers with the same base, the exponent of the result will be equal to the difference of the exponents.

$\frac{5^4}{5^3}=5^{4-3}=5^1$

$\frac{7^{2X}}{7^X}=7^{2X-X}=7^X$

$\frac{X^7}{X^5}=X^{7-5}=X^2$

Let's look at the following example of power of a power

$\left(a^n\right)^m=a^{n\cdot m}$

When we come across a power of a power, the result will be the multiplication of those powers.

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

$\left(a^X\right)^2=a^{2X}$

$\left(a\cdot b\cdot c\right)^n=a^n\cdot b^n\cdot c^n$

$\left(2\cdot3\cdot4\right)^2=2^2\cdot3^2\cdot4^2$

$\left(X\cdot2\cdot X\right)^2=X^2\cdot2^2\cdot X^2$

$\left(X^2\cdot2\cdot y^3\right)^2=X^4\cdot2^2\cdot y^6$

$(\frac{a}{b})^n=\frac{a^n}{b^n}$

$(\frac{5}{3})^2=\frac{5^2}{3^2}$

$(\frac{X}{y})^3=\frac{X^3}{y^3}$

Let's look at the following example of a negative exponent

$a^{-n}=\frac{1}{a^n}$

$\frac{1}{a^{-n}}=a^n$

This rule is often used to get rid of negative exponents.

$5^{-2}=\frac{1}{5^2}=\frac{1}{25}$

$\frac{1}{2^{-3}}=2^3=8$

$a^0=1$

Any number raised to the power of $0$ equals $1$.

$0^n=0$

The number $0$ raised to any power (other than $0$) equals $0$.

$0^0$ = undefined

The value of the number $0$ raised to the power of $0$ is undefined.

$1^n=1$

The number $1$ raised to any power is equal to $1$.

Exercise 1: (Variables in the value of the power)

Task:

Solve the following equation:

$(A^m)^n$

$(4^X)^2$

Solution:

$(4^X)^2=4^{X\times2}$

$(a^m)n^=a^{m\times n}$

Task:

Solve the exercise:

$(X²\times3)²=\text{?}$

Solution:

$(X²\times3)²=X^{2\times2}\times3²=X^4\times9=9X^4$

Before the formula:

$(a\times b)^{m=}a^m\times b^m$

And also

$(a^m)^n=$ Power of a power

Answer:

$9X^4$

Task:

Solve the exercise:

$((7\cdot3)^2)^6+(3^{-1})^3\cdot(2^3)^4=\text{?}$

Solution:

$(7\cdot3)^{2\cdot6}+3^{-1\cdot3}\cdot2^{3\cdot4}=\text{?}$

$21^{12}+3^{-3}\cdot2^{12}=\text{?}$

Answer:

$21^{12}+3^{-3}\cdot2^{12}$

For problems like the following, you can use the formula:

$(a^m)^n=a^{m\cdot n}$

Task:

Solve the following equation:

$2^3\cdot2^4+(4^3)^2+\frac{2^5}{2^3}=$

Solution:

$2^3\cdot2^4+(4^3)^2+\frac{2^5}{2^3}=$

$2^{3+4}+4^{3\cdot2}+2^{(5-3)}=2^7+4^6+2^2$

Answer:

$2^7+4^6+2^2$

The answer is supported by a series of properties:

1. $(a^m)^n=a^{m\cdot n}$
2. $\frac{a^X}{a^Y}=a^{X-Y}$
3. $a^X\cdot a^Y=a^{X+Y}$

Task:

Which expression has the greatest value?

$10^{2},2^{4},3^{7},5^{5}$

Solution:

$10^2=100$

$2^4=16$

$3^7=2187$

$5^5=3125$

Answer:

The greatest value is $5^5$

Task:

Solve the following equation:

$((4X)^{3Y})^2$

Solution:

$(4X)^{3Y\cdot2}=4X^{6Y}$

Formula:

$(a^m)^n=a^{m\cdot n}$

Assignment:

Solve the following equation:

$(4^2)^3+(9^3)^4=\text{?}$

Solution:

$(4^2)^3+(9^3)^4=\text{?}$

$4^{2\cdot3}+9^{3\cdot4}=4^6+9^{12}$

Answer:

$4^6+9^{12}$

What are the laws of exponents?

Multiplication with the same bases, division with the same bases, and power of powers.

How is multiplication with the same bases done?

The exponents are added.

How is division with the same bases done?

The exponents are subtracted.

What is a number raised to the 0 equal to?

One, as long as the base is not zero.