Difficulty: Foundational | Prerequisites: Basic algebra (solving linear equations, factoring quadratics)
Exponent rules are the mechanical backbone of every interest, growth, and decay formula you will meet in this course. This section covers the laws of exponents and how to solve equations where the variable sits in the exponent. If you are comfortable with these, the compound-interest material that follows will feel like plugging numbers into a template. If you are not, everything downstream will be harder than it needs to be.
Exponent rules let you simplify expressions where the same base is multiplied, divided, or raised to a power. Exponential equations are solved by rewriting both sides with a common base, then setting the exponents equal. These mechanics appear every time you work with compound interest, growth models, or decay.
Base (b)
The number being multiplied by itself. In b^n, the base is b.
Think of it as the building block that gets repeated.
Exponent (n)
The number that tells you how many times to multiply the base by itself. In b^n, the exponent is n.
In simple terms, this means the exponent is the "how many times" counter.
Exponential equation
An equation in which the unknown variable appears in the exponent, e.g. 5^(x+2x) = 125.
Think of it as an equation you solve by matching bases rather than isolating x in the usual way.
Common base
A shared base that both sides of an exponential equation can be rewritten with, allowing you to set the exponents equal.
In simple terms, if you can write both sides as powers of the same number, the exponents must be equal.
Product rule: b^x · b^y = b^(x+y). When multiplying same-base terms, add the exponents.
Example: (3)^2 · (3)^3 = 3^(2+3) = 3^5
Quotient rule: b^x / b^y = b^(x-y). When dividing same-base terms, subtract the exponents.
Power rule: (b^x)^y = b^(xy). When raising a power to another power, multiply the exponents.
Example: (2^3)^2 = 2^(3·2) = 2^6 = 64
Product of bases: (ab)^x = a^x · b^x. A product raised to a power distributes the power to each factor.
Example: (6^4)(2^4) = (6·2)^4 = 12^4, or equivalently written as 10^(19) in contexts with different bases.
Quotient of bases: (a/b)^x = a^x / b^x. A fraction raised to a power distributes the power to numerator and denominator.
Equality of exponents: if b^x = b^y and b > 0, b ≠ 1, then x = y. This is the principle behind solving exponential equations.
The strategy: rewrite both sides of the equation as powers of the same base, then set the exponents equal.
Step 1: Recognise or convert each side to the same base.
Step 2: Set the exponents equal and solve the resulting algebraic equation.
Step 3: Check your solutions, especially if the algebra produced a quadratic.
Worked example from the lecture:
5^(x² + 2x) = 125
Rewrite 125 as 5^3.
Now: 5^(x² + 2x) = 5^3, so x² + 2x = 3.
Rearrange: x² + 2x - 3 = 0.
Factor: (x + 3)(x - 1) = 0.
Solutions: x = -3 or x = 1.
b^x \cdot b^y = b^{x+y}\frac{b^x}{b^y} = b^{x-y}(b^x)^y = b^{xy}(ab)^x = a^x \cdot b^x\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}b^x = b^y \implies x = y \quad (b > 0,\; b \neq 1)Exponent rules underpin every compound-interest and continuous-growth formula in finance. When a bank compounds interest quarterly, the future-value expression is built entirely from these laws. Population growth, radioactive decay, and inflation calculations all rely on the same machinery.
Students often think b^x · b^y = b^(xy). It does not. You add the exponents when multiplying same-base terms.
Students often confuse (b^x)^y with b^(x+y). The power rule says you multiply exponents: (b^x)^y = b^(xy).
When solving 5^(x² + 2x) = 125, students sometimes forget to rewrite 125 as 5^3 first and try to take logarithms immediately, which works but is unnecessary and error-prone at this stage.
Students sometimes drop one of the two solutions from a quadratic that arises from an exponential equation. Always check both roots.
⚠️ Exponent law questions appear as "simplify" or "evaluate" problems on nearly every exam. They are free marks if you know the rules.
⚠️ The common-base method for solving exponential equations is tested directly and also appears embedded inside interest-rate problems later in the chapter.
⚠️ Factoring the resulting quadratic (as in x² + 2x - 3 = 0) is a frequent sub-step. If your factoring is shaky, revise it now.
True or false: b^3 · b^4 = b^12.
False. b^3 · b^4 = b^(3+4) = b^7.
Fill in the blank: (2^5)^3 = 2^___.
True or false: if 7^x = 7^4, then x = 4.
True.
Fill in the blank: 3^8 / 3^5 = 3^___.
True or false: (ab)^3 = a^3 + b^3.
False. (ab)^3 = a^3 · b^3 (multiply, not add).
Q: Simplify (5^3)(5^7).
A: 5^(3+7) = 5^10.
Q: Evaluate (2^3)^2.
A: 2^(3·2) = 2^6 = 64.
Q: Solve 3^(x² + 4x) = 81.
A: Rewrite 81 as 3^4. Then x² + 4x = 4, so x² + 4x - 4 = 0. Apply the quadratic formula: x = (-4 ± √(16 + 16)) / 2 = (-4 ± √32) / 2 = -2 ± 2√2.
Q: Solve 2^(2x+1) = 32.
A: Rewrite 32 as 2^5. Then 2x + 1 = 5, so x = 2.
Q: Simplify (4^5 · 4^{-2}) / 4^2.
A: 4^(5 + (-2)) / 4^2 = 4^3 / 4^2 = 4^(3-2) = 4^1 = 4.
This connects to compound interest and future value because the compound-interest formula B(t) = P(1 + r/k)^(kt) is built directly from these exponent rules. Continuous compounding uses e (Euler's number), which is defined as a limit involving exponents. Logarithms, which you will meet shortly, are the inverse of exponentiation and provide an alternative method for solving these same equations.
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