Exponent laws & zero exponent

Exponent laws let you simplify expressions with powers without expanding them fully. When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. When dividing, subtract them: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. When raising a power to a power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. Any non-zero number raised to the power of zero equals 1 — this isn't random; it falls naturally out of the division rule (e.g. a³ ÷ a³ = a⁰, and anything divided by itself is 1).

Key terms

Exponent:

The small raised number showing how many times to multiply a base by itself.

Zero-exponent rule:

Any non-zero number raised to the power of 0 equals 1.

Questions

1. 1. When multiplying powers with the same base, you:

   • Add the exponents
   • Subtract the exponents
   • Multiply the exponents
   • Divide the exponents

2. 2. When dividing powers with the same base, you:

   • Add the exponents
   • Subtract the exponents
   • Multiply the exponents
   • Ignore the exponents

3. 3. Any non-zero number raised to the power of 0 equals:

   • 0
   • 1
   • The number itself
   • Undefined always

4. 4. 2³ × 2² equals:

   • 2⁵
   • 2⁶
   • 4⁵
   • 2¹

5. 5. 5⁴ ÷ 5² equals:

   • 5²
   • 5⁶
   • 5⁸
   • 5¹

6. 6. (3²)³ means:

   • Multiply the exponents: 3⁶
   • Add the exponents: 3⁵
   • Subtract the exponents: 3⁻¹
   • Keep it as 3² × 3

7. 7. 10⁰ equals:

   • 0
   • 1
   • 10
   • 100

8. 8. Calculate 2³ × 2² using exponent laws:

   • 16
   • 32
   • 64
   • 128

9. 9. Calculate 5⁴ ÷ 5² using exponent laws:

   • 5
   • 15
   • 25
   • 625

10. 10. Calculate (3²)³ using exponent laws:

    • 27
    • 81
    • 729
    • 9

11. 11. Simplify x⁵ × x³:

    • x⁸
    • x¹⁵
    • x²
    • x⁵³

12. 12. Simplify a⁷ ÷ a³:

    • a⁴
    • a¹⁰
    • a²¹
    • a¹

13. 13. Simplify (b²)⁴:

    • b⁶
    • b⁸
    • b²
    • b¹⁶

14. 14. Why does a³ ÷ a³ equal a⁰, and why must this equal 1?

    • The subtraction rule gives a⁰, and anything divided by itself is 1
    • Because zero always means the answer is zero
    • This is a special exception with no reasoning
    • Because exponents cannot be subtracted

15. 15. Simplify (2x²)³ (apply the power to both the number and the exponent):

    • 8x⁶
    • 6x⁵
    • 2x⁶
    • 8x⁵

16. 16. Simplify (a⁴ × a²) ÷ a³:

    • a³
    • a⁶
    • a⁹
    • a¹

17. 17. Why is it useful to keep numbers in index/exponent form rather than always expanding them fully?

    • It keeps very large or small numbers compact and easier to manipulate using exponent laws
    • Expanded numbers are always easier to work with
    • Exponent form has no practical use
    • Exponent laws only work on expanded numbers

18. 18. Simplify (x³)² × x:

    • x⁷
    • x⁶
    • x⁵
    • x⁹

19. 19. A scientist writes a very small measurement as 3 × 10⁻⁴ instead of 0.0003. Why might exponent notation be preferred here?

    • It is more compact and easier to compare with other very large or small measurements
    • Exponent notation is always less accurate
    • Decimals are always clearer than exponent notation
    • There is no practical reason to use exponent notation

20. 20. Simplify (y⁵ ÷ y²) × y⁰:

    • y³
    • y⁷
    • y⁰
    • y⁵

21. 21. Understanding exponent laws mainly helps you to:

    • Simplify expressions with powers efficiently and correctly
    • Avoid ever using powers in calculations
    • Always expand every power fully by hand
    • Ignore the base of a power entirely