Math Resource - Laws of Exponents (Indices)

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Math Resource - Laws of Exponents

Exponents (also called Powers or Indices) are shorthand for repeated multiplication. In Afrikaans, these are known as Magte or Eksponente.

📏 The 7 Core Rules (Wette van Eksponente)

Rule Name	Mathematical Law	Example
1. Product Rule (Multiplication)	$a^m\times a^n=a^{m+n}$	$2^3\times 2^4=2^7$
2. Quotient Rule (Division)	$a^m÷a^n=a^{m-n}$	$5^6÷5^2=5^4$
3. Power of a Power	$(a^m{)}^n=a^{m\times n}$	$(3^2{)}^4=3^8$
4. Power of a Product	$(ab{)}^n=a^n\times b^n$	$(2x{)}^3=8x^3$
5. Zero Exponent	$a^0=1$	$100^0=1$
6. Negative Exponent	$a^{-n}=\frac{1}{a^n}$	$4^{-2}=\frac{1}{16}$
7. Fractional Exponent	$a^{1/n}=\sqrt[n]{a}$	$9^{1/2}=\sqrt{9}=3$

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💡 Key Concepts to Remember

The “Same Base” Rule (Dieselfde Grondtal)

Rules 1 and 2 only work if the big numbers (the bases) are the same.

• Correct: $x^2\times x^3=x^5$
• Incorrect: $x^2\times y^3\ne (xy{)}^5$ (You cannot add these exponents!)

The Zero Rule ($a^0=1$)

Any non-zero number raised to the power of zero is always 1.

• Why? Because $a^n÷a^n=1$, and according to the Quotient Rule, $a^n÷a^n=a^{n-n}=a^0$. Therefore, $a^0=1$.

Negative Exponents (Negatiewe Eksponente)

Think of a negative exponent as a “ticket” to move to the other side of the fraction line.

• If it’s on top and negative, move it to the bottom to make it positive: $x^{-3}=\frac{1}{x^3}$
• If it’s on the bottom and negative, move it to the top: $\frac{1}{x^{-2}}=x^2$

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📝 Practice Example

Problem: Simplify $\frac{(x^3{)}^2\times x^4}{x^5}$

1. Power of a Power first: $(x^3{)}^2=x^{3\times 2}=x^6$
2. Product Rule (Top): $x^6\times x^4=x^{6+4}=x^{10}$
3. Quotient Rule: $x^{10}÷x^5=x^{10-5}=x^5$

Final Answer: $x^5$

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Note: This resource is intended to simplify the transition between mathematical terminologies.