Exponents & Scientific Notation — Challenge
An exponent counts how many times you multiply, not what you multiply by. That one idea explains every rule.
What these problems cover
These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Base — The number being multiplied. In 53 the base is 5.
- Exponent — How many times the base is used as a factor. In 53 the exponent is 3.
- Power — The whole expression, 53, and also its value, 125.
- Scientific notation — A number written as a × 10n, where 1 ≤ a < 10.
- Standard form — The ordinary way of writing a number, like 47,000.
Problem 1
What is 25?
- 10
- 16
- 25
- 32
- 64
Worked answer
32
The exponent counts how many factors of 2 are multiplied, not what 2 is multiplied by.
2· 2· 2· 2· 2 = 32. The answer 10 would come from 2· 5.
Problem 2
What is 34· 32?
- 243
- 729
- 2187
- 6561
- 9
Worked answer
729
Multiplying powers of the same base adds the exponents: 34· 32 = 36.
36 = 729. Multiplying the exponents instead would give 38, which is 6561.
Problem 3
What is (23)2÷ 24?
- 2
- 4
- 8
- 16
- 32
Worked answer
4
A power raised to a power multiplies the exponents: (23)2 = 26.
Dividing powers of the same base subtracts the exponents: 26÷ 24 = 22.
22 = 4.
Problem 4
What is 2-3?
- 18
- -8
- 16
- -18
- 8
Worked answer
18
A negative exponent means the reciprocal of the positive power, not a negative answer: 2-3 = 123.
123 = 18, which is positive.
Problem 5
What is 103?
- 30
- 100
- 300
- 1000
- 10000
Worked answer
1000
A power of ten has as many zeros as the exponent.
103 = 10· 10· 10 = 1000, which is a 1 followed by three zeros.
Problem 6
What is 50+51?
- 1
- 5
- 6
- 10
- 25
Worked answer
6
Any non-zero number to the power 0 is 1, so 50 = 1.
Any number to the power 1 is itself, so 51 = 5.
1+5 = 6.
Problem 7
What is the value of x5x2 when x=3?
- 9
- 15
- 27
- 81
- 243
Worked answer
27
Simplify before substituting: dividing powers of the same base subtracts the exponents, so x5x2 = x3.
With x=3 that is 33 = 27.
Working out 2439 gives the same answer, but simplifying first is less work.
Problem 8
What is 42?
- 8
- 12
- 16
- 18
- 42
Worked answer
16
Squaring means multiplying the number by itself.
4· 4 = 16. Doubling would give 8, which is the common slip.
Problem 9
What is 32· 23?
- 36
- 54
- 72
- 216
- 1296
Worked answer
72
The bases differ, so the exponents cannot be combined. Work out each power separately.
32 = 9 and 23 = 8, so the product is 9· 8 = 72.
Problem 10
What is 210÷ 27?
- 2
- 4
- 8
- 16
- 128
Worked answer
8
Dividing powers of the same base subtracts the exponents: 210÷ 27 = 23.
23 = 8. Note that the answer does not depend on working out 1024 and 128 first.