Exponents: Laws, Negative Powers & Equations

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Exponents are shorthand for repeated multiplication: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$. The base is the number being multiplied; the exponent tells you how many times. In Grade 10, you learn the laws that make working with exponents fast and efficient.

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The Golden Rule

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Exponent laws only work when the BASES are the same. You can simplify $2^3 \times 2^4$ but NOT $2^3 \times 3^4$.

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The Five Core Laws

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Law	Rule	Example
Product (same base)	$x^a \cdot x^b = x^{a+b}$	$2^3 \cdot 2^4 = 2^7 = 128$
Quotient (same base)	$\frac{x^a}{x^b} = x^{a-b}$	$\frac{3^5}{3^2} = 3^3 = 27$
Power of a power	$(x^a)^b = x^{ab}$	$(2^3)^4 = 2^{12}$
Power of a product	$(xy)^a = x^a y^a$	$(3x)^2 = 9x^2$
Power of a quotient	$\left(\frac{x}{y}\right)^a = \frac{x^a}{y^a}$	$\left(\frac{2}{3}\right)^3 = \frac{8}{27}$

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Special Cases

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Zero exponent

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

Why? Because $\frac{x^n}{x^n} = x^{n-n} = x^0$, and anything divided by itself is 1.

Negative exponent

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

A negative exponent means “take the reciprocal”:

• $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
• $\frac{1}{x^{-2}} = x^2$ (the negative “flips” back)

Fractional exponents (preview of Grade 11)

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$$x^{\frac{1}{n}} = \sqrt[n]{x} \qquad x^{\frac{m}{n}} = \sqrt[n]{x^m}$$

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Solving Exponential Equations

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Strategy: Get the same base on both sides, then set the exponents equal.

Example: Solve $2^{x+1} = 16$

1. Rewrite 16 as a power of 2: $2^{x+1} = 2^4$
2. Same bases → exponents are equal: $x + 1 = 4$
3. $x = 3$

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Deep Dive

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• Exponent Laws, Simplification & Equations — full worked examples for each law, simplifying complex expressions, and solving exponential equations

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🚨 Common Mistakes

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1. Adding exponents when multiplying different bases: $2^3 \times 3^2 \neq 6^5$. Laws only work for the same base.
2. Distributing exponents over addition: $(x + y)^2 \neq x^2 + y^2$. You must expand the brackets properly.
3. Negative exponent confusion: $2^{-3} = \frac{1}{8}$, NOT $-8$.
4. Zero exponent: $0^0$ is undefined. But $5^0 = 1$, $(-3)^0 = 1$, $(100x)^0 = 1$.

🔗 Related Grade 10 topics:

• Algebra — exponent laws are used constantly when expanding and factorising
• Equations — exponential equations are a key equation type

📌 Where this leads in Grade 11: Exponents & Surds — fractional exponents, surd laws, and rationalising

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⏮️ Algebra | 🏠 Back to Grade 10 | ⏭️ Number Patterns