1. Repeated multiplication

aⁿ = the product of the real number a multiplied by itself n times.

• a = base (the repeated factor) , n = exponent (number of times the base is repeated).
• $\sqrt{3}$ $\times$ $\sqrt{3}$ $\times$ $\sqrt{3}$ $\times$ $\sqrt{3}$ = ($\sqrt{3}$)⁴ — read "$\sqrt{3}$ raised to the power of 4".

Sign rule :

• (-a)ⁿ = aⁿ if n is EVEN. Example : (-$\sqrt{5}$)⁴ = ($\sqrt{5}$)⁴ = 25.
• (-a)ⁿ = -aⁿ if n is ODD. Example : (-$\sqrt{5}$)³ = -($\sqrt{5}$)³ = -5$\sqrt{5}$.

2. Zero exponent

Any real number (not equal to zero) raised to the power 0 equals 1 :

• a⁰ = 1 (a $\neq$ 0). Examples : 3⁰ = 1 , ($\sqrt{7}$)⁰ = 1 , (-2$\sqrt[3]{4}$)⁰ = 1.

3. Negative exponent

If a $\neq$ 0 and n is a positive integer :

• a⁻ⁿ = 1/aⁿ and aⁿ = 1/a⁻ⁿ
• Examples : 5⁻² = 1/25 ; 1/3⁻² = 9 ; ($\sqrt{7}$)⁻² = 1/7 ; ($\sqrt{2}$)⁻³ = ($\sqrt{2}$)³ = 2$\sqrt{2}$.

4. The LAWS of exponents (m , n integers, bases $\neq$ 0, avoid 0⁰)

5. Powers of 10 (standard form help)

• 10 = 10¹ , 100 = 10² , 1000 = 10³
• 0.1 = 10⁻¹ , 0.01 = 10⁻² , 0.001 = 10⁻³

6. Combining radicals with exponent laws

Everything works with bases like $\sqrt{2}$ , $\sqrt{3}$ , $\sqrt[3]{4}$ , etc. Example :

• ($\sqrt{2}$)² $\times$ ($\sqrt{2}$)⁴ = ($\sqrt{2}$)⁶ = 8
• ($\sqrt{7}$)⁵ $\times$ ($\sqrt{7}$)⁻² $\times$ ($\sqrt{7}$)⁻¹ = ($\sqrt{7}$)² = 7
• ($\sqrt{5}$)⁻⁴ ÷ ($\sqrt{5}$)⁻⁶ = ($\sqrt{5}$)² = 5

7. WORKED EXAMPLES

Example 1 : Simplify (x³ $\times$ x⁻² $\times$ x⁻¹).

• x³ $\times$ x⁻² $\times$ x⁻¹ = x³⁻²⁻¹ = x⁰ = 1.

Example 2 : Simplify ($\sqrt{3}$)⁴ $\times$ 27 / (($\sqrt{3}$)⁵ $\times$ ($\sqrt{3}$)⁴).

• Note 27 = ($\sqrt{3}$)⁶. Numerator ($\sqrt{3}$)⁴ $\times$ ($\sqrt{3}$)⁶ = ($\sqrt{3}$)¹⁰ ; denominator ($\sqrt{3}$)⁹ ; result ($\sqrt{3}$)¹ = $\sqrt{3}$.

Example 3 : If 3ˣ = 5, find 3ˣ⁺¹.

• 3ˣ⁺¹ = 3ˣ $\times$ 3¹ = 5 $\times$ 3 = 15.

Example 4 : Simplify (36)ⁿ $\times$ (5)²ⁿ / (30)²ⁿ.

• 36 = 6² $\rightarrow$ 6²ⁿ ; 5²ⁿ ; 30 = 6 $\times$ 5 $\rightarrow$ 30²ⁿ = 6²ⁿ $\times$ 5²ⁿ $\rightarrow$ result = 1.