Year 9AC9M9A01

Exponential Equations

Apply the exponent laws to numerical expressions with integer exponents and extend to variables.

Get both sides to the same base, then the exponents must be equal.

Worked example

e.g.

8 \times 27^{x+5} = 3 \times 3^{4x-4} \times 24

Change all bases into their prime factors. Keep brackets around the exponents. $2^{3} \times \left(3^{3}\right)^{x+5} = 3 \times 3^{4x-4} \times 2^{3} \times 3$
Distribute all exponents into the factors $2^{3} \times 3^{3x+15} = 3 \times 3^{4x-4} \times 2^{3} \times 3$
If unlike bases occur, divide both sides to get rid of them $\frac{2^{3} \times 3^{3x+15}}{2^{3}} = \frac{3 \times 3^{4x-4} \times 2^{3} \times 3}{2^{3}}$
Simplify bases on both sides $3^{3x+15} = 3^{4x-4+1+1}$
Drop the like bases $3x + 15 = 4x - 2$
Solve $x = 17$

2^{3x} = 16

Answer

x = \tfrac{4}{3}

5^{-x} = 5^{4}

Answer

x = -4

6^{x+2} \times 5 = 36 \times 5

Answer

x = 0

3^{2x} \times 3^{-3x} = 1

Answer

x = 0

\frac{2^{-2x}}{32} = 2^{x}

Answer

x = -\tfrac{5}{3}

6^{3x-1} = 1

Answer

x = \tfrac{1}{3}

64 \times 4^{3x} = 16

Answer

x = -\tfrac{1}{3}

2^{3x+3} \times 16 = 2^{x} \times 64

Answer

x = -\tfrac{1}{2}