Find the value of x if (27/8)^(x+7) = (4/9)^(-3x)

Equation iliyotolewa ni:

$$\left(\frac{27}{8}\right)^{x+7} = \left(\frac{4}{9}\right)^{-3x}$$

Tunaweza kuandika $27 = 3^3$, $8 = 2^3$, $4 = 2^2$, na $9 = 3^2$.
Hivyo, equation inakuwa:

$$\left(\frac{3^3}{2^3}\right)^{x+7} = \left(\frac{2^2}{3^2}\right)^{-3x}$$ $$\left(\left(\frac{3}{2}\right)^3\right)^{x+7} = \left(\left(\frac{2}{3}\right)^2\right)^{-3x}$$

$$\left(\frac{3}{2}\right)^{3(x+7)} = \left(\frac{2}{3}\right)^{2(-3x)}$$ $$\left(\frac{3}{2}\right)^{3x+21} = \left(\frac{2}{3}\right)^{-6x}$$

Tunajua kuwa $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$. Hivyo, tunaweza kubadilisha base ya upande wa kulia:

$$\left(\frac{2}{3}\right)^{-6x} = \left(\frac{3}{2}\right)^{6x}$$

Sasa equation inakuwa:

$$\left(\frac{3}{2}\right)^{3x+21} = \left(\frac{3}{2}\right)^{6x}$$

Kwa kuwa bases ni sawa, exponents lazima ziwe sawa:

$$3x+21 = 6x$$

Toa $3x$ kutoka pande zote mbili:

$$21 = 6x - 3x$$ $$21 = 3x$$

Gawanya kwa 3:

$$x = \frac{21}{3}$$ $$x = 7$$

Thamani ya $x$ ni:

$$7$$