Powers

Question 1

Calculate

2²

Answer 1

4

Question 2

Calculate

2⁴

Answer 2

16

Question 3

Calculate

2⁵

Answer 3

32

Question 4

Calculate

2⁷

Answer 4

128

Question 5

Calculate

2⁸

Answer 5

256

Question 6

Calculate

2⁹

Answer 6

512

Question 7

Calculate

9²

Answer 7

81

Question 8

Calculate

11²

Answer 8

121

Question 9

Calculate

12²

Answer 9

144

Question 10

Calculate

13²

Answer 10

169

Question 11

Calculate

14²

Answer 11

196

Question 12

Calculate

15²

Answer 12

225

Question 13

Calculate

16²

Answer 13

256

Question 14

Calculate

17²

Answer 14

289

Question 15

Calculate

18²

Answer 15

324

Question 16

Calculate

19²

Answer 16

361

Question 17

Calculate

10³

Answer 17

1000

Question 18

Calculate

3³

Answer 18

27

Question 19

Calculate

8²

Answer 19

64

Question 20

Write down as a multiplication

4⁷

Answer 20

4 · 4 · 4 · 4 · 4 · 4 · 4

Question 21

Write down as a product of powers

2 · 3 · 2 · 2 3 · 3 · 2 · 3

Answer 21

2⁴ · 3⁴

Question 22

Write down as a product of powers

5 · 3 · 2 · 2 5 · 5 · 2 · 3

Answer 22

2³ · 3² · 5³

Question 23

Calculate

[(3²)⁵]⁰

Answer 23

1

Question 24

Calculate

(0⁶)²

Answer 24

0

Question 25

Calculate

1²⁰+20¹

Answer 25

21

Question 26

Calculate

10³ : 10

Answer 26

100

Question 27

Calculate

2⁵ = and 5² =

What does this example show?

Answer 27

2⁵ = 32 and 5² = 25

It shows that the commutative property does not hold for powers!

Question 28

Is this equivalence true?

(3 + 2)² ?=? 3²+2²

What does that prove?

Answer 28

No it is not:

(3+2)² = 5² = 25
while
3²+2² = 9+4 = 13

It shows that the distributive property does not hold for powers.

Question 29

Can you find the value of

0⁰ ?

Answer 29

No, it is an undefined expression.

Question 30

What is the meaning of

aⁿ

where a and n are positive integers?

Answer 30

The definition of power says that

aⁿ = a · a · a · a · … · a (n times)

Question 31

What is the difference between

4 times 5

and

4, mutliplied 5 times ?

Answer 31

The first is the multiplication:

4 · 5 = 20

The second expression is a power:

4 · 4 · 4 · 4 · 4 = 1024

Question 32

Find the values of n,m and k so that

2ⁿ = 4^m = 16^k = 256

Answer 32

n=8, m=4, k=2

in fact

2⁸ = 4⁴ = 16² = 256

Question 33

Find the exponent n

3ⁿ = 27

Answer 33

n=3

Question 34

Find the exponent n

10ⁿ = 10’000

Answer 34

n=4

Question 35

Find the exponent n

13ⁿ=169

Answer 35

n=2

Question 36

Find the exponent n

225ⁿ = 1

Answer 36

n=0

Question 37

Is it possible to find a whole number n so that

3ⁿ = 0

Answer 37

No, all the powers of three are greater than zero. The smallest possible power is three is 3⁰ which is exctually equal to 1.

Question 38

Find the base b

b⁴ = 16

Answer 38

b=2

Question 39

Find the base b

b¹ = 17

Answer 39

b=17

Question 40

Find the base b

b⁶=1

Answer 40

b=1

Question 41

Find the misterious number x

x^x = x

Answer 41

x=1

Question 42

Find the misterious number x

x^x = 4

Answer 42

x=2

Question 43

Find two numbers, a and b, so that

a^b=b^a

Answer 43

This problem has many solutions:

1) if you choose a=b (for example a=5 and b=5) this equality is always true
2) a=2 and b=4 or viceversa a=4 and b=2

Question 44

Is it possible to find two DIFFERENT whole numbers x and y so that

x² = y²

Answer 44

No, each square gives a different result. If x is smaller than y, for instance, then x² will necessarily be smaller than y², and viceversa…