When multiplying powers with the same base, keep the base and add the exponents.

$x^m \times x^n = x^{m+n}$

Worked Example 1

Simplify $7^6 \times 7^8$

Solution

Since the bases are the same (both are 7), we add the exponents:

$7^6 \times 7^8 = 7^{6+8} = 7^{14}$

When dividing powers with the same base, keep the base and subtract the exponent of the divisor from the exponent of the dividend.

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

Worked Example 2

Simplify $\frac{3^{20}}{3^{12}}$

Solution

Subtract the exponents:

$\frac{3^{20}}{3^{12}} = 3^{20-12} = 3^8$

When raising a power to another power, keep the base and multiply the exponents.

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

Worked Example 3

Simplify $(3^6)^3$

Solution

Multiply the exponents:

$(3^6)^3 = 3^{6 \times 3} = 3^{18}$

Any non-zero number raised to the power of zero equals 1.

$x^0 = 1 \quad \text{(where } x \neq 0\text{)}$

Worked Example 4

Simplify $\frac{7^3}{7^3}$

Solution

Using the division law:

$\frac{7^3}{7^3} = 7^{3-3} = 7^0$

But we also know that any number divided by itself equals 1. Therefore:

$7^0 = 1$

A negative exponent means we take the reciprocal (one over) of the base and change the exponent to positive.

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

Worked Example 5

Express $4^{-3}$ using positive exponents

Solution

$4^{-3} = \frac{1}{4^3} = \frac{1}{64}$

Worked Example 6

Simplify $\frac{x^5}{x^7}$ and write with positive exponents

Solution

$\frac{x^5}{x^7} = x^{5-7} = x^{-2} = \frac{1}{x^2}$

When solving equations with exponents, remember:

• If bases are the same, equate the exponents: if $2^x = 2^5$, then $x = 5$
• Express both sides as powers of the same base first

Worked Example 7

Find the value of $n$ in $2^{n+1} = 64$

Solution

Express 64 as a power of 2: $64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6$

So: $2^{n+1} = 2^6$

Equating exponents: $n + 1 = 6$ $n = 5$

In Tanzania, exponents are used when calculating interest on savings or loans. For example, if a mobile money lending app applies a monthly interest rate where the amount owed grows by a factor of 1.02 each month, the amount after $n$ months can be expressed as $P \times (1.02)^n$, where $P$ is the principal amount. Using the laws of exponents, we can simplify and calculate the final amount quickly without multiplying 1.02 by itself many times.