Solve Exponential Equations

An exponential equation is an equation in which a variable appears in the exponent. For example, $$2^x = 8$$ .

Convert 64 to a power of 4: $$64 = 4^3$$ . Therefore, $$p + 2 = 3$$ , leading to $$p = 1$$ .

The base is the constant that is raised to a power. In $$b^x$$ , 'b' is the base.

Convert 8 to base 2: $$8 = 2^3$$ . Thus, $$2^3 = 2^{5x+7}$$ , leading to $$3 = 5x + 7$$ .

Convert $$\frac{1}{16}$$ to a power of 4: $$\frac{1}{16} = 4^{-2}$$ . Therefore, $$3b - 3 = -2$$ , leading to $$b = \frac{1}{3}$$ .

$$6^{-2} = \frac{1}{6^2} = \frac{1}{36}$$ .

Since the bases are the same, set the exponents equal: $$-m = 3m - 3$$ . Solving gives $$m = \frac{3}{4}$$ .

This property allows you to combine exponents with the same base by adding their powers.

This property allows you to simplify fractions with the same base by subtracting the exponents.

This property allows you to simplify powers of powers by multiplying the exponents.