# Converting bases to base ten (How to)

When converting bases to base ten, it has to undergo either of these two methods; a successive multiplication method or by power Expansion method. This simply means the reverse of the very one used when converting from base ten to another bases.

## Method 1 (Converting bases): Successive multiplication.

This method of converting bases is the reverse of the consecutive division done when converting from base ten to another bases.

## Method 2 (Converting bases): Power Expansion.

converting bases to base ten: This is where the use of place values comes in. We multiply out the number base using their place values.

Example 1:

Convert 101101 base two to base ten.

## Using consecutive/successive multiplication method for Converting bases.

converting bases to base ten:

Solution;

Integer given = 101101

Base given = base two

Base required = base ten

First step is to multiply the given base by the first digit (1)01101 of the integer given and adding the product to the nearest digit 1(0)1101.

So, 2 x 1 + 0 = 2 + 0 = 2

Second step is to multiply the result gotten from step one by the base given and add the third digit 10(1)101 of the integer to the product gotten.

2 x 2 + 1 = 4 + 1 = 5

Third step is to multiply the result gotten from the second step by the base given and add the forth digit 101(1)01 of the integer to the product gotten.

2 x 5 + 1 = 10 + 1 = 11

Forth step is to multiply the result gotten from third step by the base given and add the fifth digit 1011(0)1 of the integer to the product gotten.

2 x 11 + 0 = 22 + 0 = 22

Fifth step is to multiply the result gotten from forth step by the base given and add the final digit 10110(1) of the integer to the product gotten.

2 x 22 + 1 = 44 + 1 = 45 base ten.

Note the number of digits that make up the integer determine the number of steps to be taken.

## Using Power Expansion to solve example 1.

101101 base two

= 1 x 2^5 + 0 x 2 ⁴ + 1 x 2³ + 1 x 2² + 0 x 2¹ + 1 x 2°

= 1 x 32 + 0 x 16 + 1 x 8 + 1 x 4 + 0 x 2 + 1 x 1

= 32 + 0 + 8 + 4 + 0 + 1 = 45 base ten.

Students must be wondering how we got all those raised to powers.

Taking good look at this 101101

We devised this formula Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴…   to help us know the power that will be raised to the base at every digit of the integer.

Where

D = Digit

b = base

n = Total number of the digits.

### Example 2.

Change 825 base nine to base ten.

Using successive multiplication method of converting bases to base ten.

Solution:

825 base nine

9 x 8 + 2 = 74

##### Second step:

9 x 74 + 5 = 671 base ten.

Using Power Expansion:

Solution:

825 base nine

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 8 x 9³–¹ + 2 x 9³–² + 5 x 9³–³

= 8 x 9² + 2 x 9¹ + 5 x 9° = 8 x 81 + 2 x 9 + 5 x 1

= 648 + 18 + 5  = 671 base ten.

## QUESTIONS AND SOLUTIONS

### Q1. 132 base four

Solution

Method 1.

4 x 1 + 3 = 7

4 x 7 + 2 = 30 base ten

Method 2.

132 base four

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 1 x 4² + 3 x 4¹ + 2 x 4°

= 16 + 12 + 2  = 30 base ten.

### Q2. 1026 base seven

Method 1

7 x 1 + 0 = 7

7 x 7 + 2 = 51

7 x 51 + 6 = 363 base ten

Method 2.

1026 base seven

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴

= 1 x 7³ + 0 x 7² + 2 x 7¹ + 6 x 7°

= 343 + 0 + 14 + 6 = 363 base ten.

### Q3. 49 base twelve

Method 1.

12 x 4 + 9 = 57 base ten

Method 2.

49 base twelve

= Dbⁿ–¹ + Dbⁿ–² = 4 x 12¹ + 9 x 12°

= 48 + 9 = 57 base ten.

### Q4. 324 base six

Method 1.

6 x 3 + 2 = 20

6 x 20 + 4 = 124 base ten.

Method 2.

324 base six

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 3 x 6² + 2 x 6¹ + 4 x 6°

= 108 + 12 + 4 = 124 base ten.

### Q5. 110110 base two

Method 1.

2 x 1 + 1 = 3

2 x 3 + 0 = 6

2 x 6 + 1 = 13

2 x 13 + 1 = 27

2 x 27 + 0 = 54 base ten.

Method 2.

110110 base two

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴ + Dbⁿ–5 + Dbⁿ–6

= 1 x 2^5 + 1 x 2⁴ + 0 x 2³ + 1 x 2² + 1 x 2¹ + 0 x 2°

= 32 + 16 + 0 + 4 + 2 + 0 = 54 base ten.

### Q6. 3120 base five

Method 1.

5 x 3 + 1 = 16

5 x 16 + 2 = 82

5 x 82 + 0 = 410 base ten.

Method 2.

3120 base five

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴

= 3 x 5³ + 1 x 5² + 2 x 5¹ + 0 x 5°

= 375 + 25 +10 + 0 = 410 base ten.

### Q7. 1000 base two.

Method 1.

2 x 1 + 0 = 2

2 x 2 + 0 = 4

2 x 4 + 0 = 8 base ten.

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴

= 1 x 2³ + 0 x 2² + 0 x 2¹ + 0 x 2°

= 8 + 0 + 0 + 0 = 8 base ten.

### Q8. 34 base six.

Method 1.

6 x 3 + 4 = 22 base ten.

Method 2.

= Dbⁿ–¹ + Dbⁿ–² = 3 x 6¹ + 4 x 6°

= 18 + 4 = 22 base ten.

### Q9. 112 base three.

Method 1.

3 x 1 + 1 = 4

3 x 4 + 2 = 14 base ten.

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 1 x 3² + 1 x 3¹ + 2 x 3°

= 9 + 3 + 2 = 14 base ten.

### Q10. 163 base eight.

Method 1.

8 x 1 + 6 = 14

8 x 14 + 3 = 115 base ten.

Method 2

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 1 x 8² + 6 x 8¹ + 3 x 8°

= 64 + 46 + 3 = 115 base ten.

### Q11. 214 base nine

Method 1

9 x 2 + 1 = 19

9 x 19 + 4 = 175 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 2 x 9² + 1 x 9¹ + 4 x 9°

= 162 + 9 + 4 = 175 base ten

### Q12. 1407 base eight

Method 1.

8 x 1 + 4 = 12

8 x 12 + 0 = 96

8 x 96 + 7 = 775 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴

= 1 x 8³ + 4 x 8² + 0 x 8¹ +  7 x 8°

= 512 + 256 + 0 + 7 = 775 base ten

### Q13. 230 base five

Method 1

5 x 2 + 3 = 13

5 x 13 + 0 = 65 base ten

Method 2

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 2 x 5² + 3 x 5¹ + 0 x 5°

= 50 + 15 + 0 = 65 base ten

### Q14. 1104 base six

Method 1

6 x 1 + 1 = 7

6 x 7 + 0 = 42

6 x 42 + 4 = 256 base ten

Method 2

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴

= 1 x 6³ + 1 x 6² + 0 x 6¹ + 4 x 6°

= 216 + 36 + 0 + 4 = 256 base ten

### Q15. 101101 base two

Method 1

2 x 1 + 0 = 2

2 x 2 + 1 = 5

2 x 5 + 1 = 11

2 x 11 + 0 = 22

2 x 22 + 1 = 45 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴ + Dbⁿ–5 + Dbⁿ–6

= 1 x 2^5 + 0 x 2⁴ + 1 x 2³ + 1 x 2² + 0 x 2¹ + 1 x 2°

= 32 + 0 + 8 + 4 + 0 + 1 = 45 base ten

### Q16. 10011 base two

Method 1.

2 x 1 + 0 = 2

2 x 2 + 0 = 4

2 x 4 + 1 = 9

2 x 9 + 1 = 19 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴ + Dbⁿ–5

= 1 x 2⁴ + 0 x 2³ + 0 x 2² + 1 x 2¹ + 1 x 2°

= 16 + 0 + 0 + 2 + 1 = 19 base ten

### Q17. 2130 base four

Method 1

4 x 2 + 1 = 9

4 x 9 + 3 = 39

4 x 39 + 0 = 156 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴

= 2 x 4³ + 1 x 4² + 3 x 4¹ + 0 x 4°

= 128 + 16 + 12 + 0 = 156 base ten

### Q18. 56 base seven.

Method 1

7 x 5 + 6 = 41 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–²

= 5 x 7¹ + 6 x 7°

= 35 + 6

= 41 base ten

### Q19. 841 base twelve

Method 1

12 x 8 + 4 = 100

12 x 100 + 1 = 1201 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 8 x 12²  + 4 x 12¹ + 1 x 12°

= 1152 + 48 + 1 = 1201 base ten

### Q20. 3241 base six

Method 1

6 x 3 + 2 = 20

6 x 20 + 4 = 124

6 x 124 + 1 = 745 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴

= 3 x 6³ + 2 x 6² + 4 x 6¹ + 1 x 6°

= 648 + 72 + 24 + 1 = 745 base ten.