Powers and Roots

Study Notes

Powers and roots are fundamental concepts in mathematics that involve multiplying a number by itself and finding the inverse operation, respectively.

• Power — A number multiplied by itself several times. Example: $2 \times 2 \times 2 \times 2 = 2^4$
• Multiplying Powers — When multiplying powers with the same base, add the exponents. Example: $2^5 \times 2^3 = 2^{5+3} = 2^8$
• Dividing Powers — When dividing powers with the same base, subtract the exponents. Example: $2^5 \div 2^3 = 2^{5-3} = 2^2$
• Power of a Power — Multiply the exponents when raising a power to another power. Example: $(2^3)^2 = 2^{3\times2} = 2^6$
• Negative Power — A negative exponent can be converted to a positive by taking the reciprocal. Example: $10^{-3} = \frac{1}{10^3}$
• Root — The inverse operation of a power, represented by the root symbol. Example: $\sqrt{4} = 4^{1/2} = 2$
• Higher Order Roots — Indicated by symbols like $\sqrt[3]{}, \sqrt[4]{}, \sqrt[5]{}$. Example: $\sqrt[3]{27} = 27^{1/3} = 3$

Exam Tips

Key Definitions to Remember

• Power: A number multiplied by itself several times
• Root: The inverse operation of a power

Common Confusions

• Confusing multiplying powers with adding them
• Misunderstanding negative exponents as negative numbers

Typical Exam Questions

• What is $2^3 \times 2^2$? $2^5$
• How do you simplify $\frac{2^5}{2^2}$? $2^3$
• What is $\sqrt{16}$? $4$

What Examiners Usually Test

• Understanding and applying the rules of powers
• Calculating roots and powers using a calculator
• Simplifying expressions involving powers and roots