2. Division Law: Same Base ($\frac{a^m}{a^n}$)

When dividing powers with the same base, you subtract the exponents.

Formula: $\frac{a^m}{a^n} = a^{m-n}$ (where $m > n$)

Example: $\frac{2^8}{2^5}$

Solution:

After cancelling 5 factors of 2 from the numerator and denominator:

Using the formula:

2.1 Application: Comparing Exponential Values

Example A: Simple Comparison

Example: Which one is greater: $2^8$ or $8^2$?

Solution:

• $2^8 = 256$

• $8^2 = 64$Since $256 > 64$, thus $2^8 > 8^2$.

Example B: Scientific Notation Comparison

Example: Compare $2.7 \times 10^{12}$ and $1.5 \times 10^8$.

Solution:

To compare numbers in scientific notation, it is easiest to express them with the same power of 10.

Let $a = 2.7 \times 10^{12}$ and $b = 1.5 \times 10^8$.

We rewrite $a$:

Since $27000 > 1.5$, we can conclude that $2.7 \times 10^{12} > 1.5 \times 10^8$.

2.2 Application: Prime Factorization

The laws of exponents are based on representing numbers as products of their prime factors.

Example: Express $432$ as a product of powers of prime factors.

Solution:

The prime numbers are $2, 3, 5, 7, 11…$.

Using the division method :

In power notation:

2.3. Application: Simplifying Complex Expressions

Example: Simplify: (i) $2^4 \times 3^2$, (ii) $(-3)^3 \times (-5)^3$, and (iii) $2^5 \times 5^4$.

• (i) $2^4 \times 3^2$
  $$16 \times 9 = 144 \text{}$$
• (ii) $(-3)^3 \times (-5)^3$
  $$(-3) \times (-3) \times (-3) \times (-5) \times (-5) \times (-5) \text{}$$
  $$\text{Based on the provided solution steps: } -9 \times 125 = -1125 \text{}$$

   

  (Note: Mathematically, $(-3)^3$ is $-27$ and $(-5)^3$ is $-125$. The product should be $3375$.)

• (iii) $2^5 \times 5^4$
  $$2^5 \times 5^4 = 2 \times 2^4 \times 5^4 \text{ (Using the commutative property)}$$
  $$= 2 \times 10^4 \text{}$$
  $$= 2 \times 10000 = 20000 \text{}$$