Product means multiplication.

Change the numbers to base 10 and multiply and change it back to base seven.

124_seven = 1 x 7² + 2 x 7¹ + 4 x 7⁰

124_seven = 67

23_seven = 2 x 7¹ + 3 x 7⁰

23_seven = 17

124_seven x 23_seven = 67 x 17

124_seven x 23_seven = 1139

1139 base 10 to base 7

$$\begin{array}{cc}\mathrm{}& 1139\\ 7& \mathrm{162\; r\; 5}\\ 7& \mathrm{23\; r\; 1}\\ 7& \mathrm{3\; r\; 2}\\ 7& \mathrm{0\; r\; 3}\end{array}$$

You list the remainders from the bottom to the top.

1139_ten = 3215_seven

Method 1

Simplifying in base six

Solving for the addition

Note: 7 in base six is 11

Solving for the difference

Method II

Change the base six to base 10 and simplify and change the result back to base six

141_six = 1 x 6² + 4 x 6¹ + 1 x 6⁰

141_six = 36 + 24 + 1

141_six = 61

233_six = 2 x 6² + 3 x 6¹ + 3 x 6⁰

233_six = 72 + 18 + 3

233_six = 93

102_six = 1 x 6² + 0 x 6¹ + 2 x 6⁰

102_six = 36 + 0 + 2

102_six = 38

141_six + 233_six - 102_six = 61 + 93 - 38

141_six + 233_six - 102_six = 116

Change the base 10 to six

116 base 10 to base 6

$$\begin{array}{cc}\mathrm{}& 116\\ 6& \mathrm{19\; r\; 2}\\ 6& \mathrm{3\; r\; 1}\\ 6& \mathrm{0\; r\; 3}\end{array}$$

You list the remainders from the bottom to the top.

116_ten = 312_six

If you convert each of them to base 10, you should get the same value since they are equal.

2 x x¹ + 3 x x⁰ = 1 x 2³ + 1 x 2² + 1 x 2¹ + 1 x 2⁰

a⁰ = 1 (Every number raised to the power 0 is 1

a¹ = a (Every number to the power 1 is the same number)

2 x x + 3 x 1 = 1 x 8 + 1 x 4 + 1 x 2 + 1 x 1
2x + 3 = 8 + 4 + 2 + 1
2x = 8 + 4 + 2 + 1 - 3
2x = 12

Divide both sides by 2

x = $\frac{12}{2}$

x = 6

Method I

A number raised to the power 2 (square) means the number multiplied by itself.

Note: 2 in base two is 10.

Method II

Change the binary (base two) to base 10 and simplify and change the result back to base two.

11_two = 1 x 2¹ + 1 x 2⁰

11_two = 1 x 2 + 1 x 1

Note: any number raised to the power 0 is 1.

11_two = 2 + 1 = 3

(11_two)² = 3² = 3 x 3 = 9

Change the 9 base 10 to base 2.

9 base 10 to base 2

$$\begin{array}{cc}\mathrm{}& 9\\ 2& \mathrm{4\; r\; 1}\\ 2& \mathrm{2\; r\; 0}\\ 2& \mathrm{1\; r\; 0}\\ 2& \mathrm{0\; r\; 1}\end{array}$$

You list the remainders from the bottom to the top.

9_ten = 1001_two

Change each of the bases to base 10.

101_two = 1 x 2² + 0 x 2¹ + 1 x 2⁰

Note: any number raised to the power 0 is 1.

101_two = 4 + 0 + 1

101_two = 5

12_y = 1 x y¹ + 2 x y⁰

12_y = y + 2

23_five = 2 x 5¹ + 3 x 5⁰

23_five = 10 + 3

23_five = 13

Substitute the values of each of the bases.

5 + (y + 2) = 13

5 + y + 2 = 13

y + 7 = 13

y = 13 - 7

y = 6

Change both bases to base 10 to determine the value of y.

23_{y = 2 x y¹ + 3 x y⁰}

Any number raised to the power 0 is 1.

23_{y = 2 x y + 3 x 1}

23_{y = 2y + 3}

1111_two = 1 x 2³ + 1 x 2² + 1 x 2¹ + 1 x 2⁰

1111_two = 8 + 4 + 2 + 1

1111_two = 15

2y + 3 = 15

2y = 15 - 3

2y = 12

Divide both sides by 2

y = $\frac{12}{2}$ = 6

Calculate the value of 123_five in base 10 and change the base 10 to base six to get it's value in base six (M).

123_five = 1 x 5² + 2 x 5¹ + 3 x 5⁰

123_five = 25 + 10 + 3 = 38

38 base 10 to base 6

$$\begin{array}{cc}\mathrm{}& 38\\ 6& \mathrm{6\; r\; 2}\\ 6& \mathrm{1\; r\; 0}\\ 6& \mathrm{0\; r\; 1}\end{array}$$

You list the remainders from the bottom to the top.

38_ten = 102_six

M = 102

10101 + 1001 = 11110

***** = 111001 - 11110

***** = 11011

Note: In base 2, when you borrow, the value is two (2).

2120₃ = 2 x 3³ + 1 x 3² + 2 x 3¹ + 0 x 3⁰

2120₃ = 2 x 27 + 1 x 9 + 2 x 3 + 0 x 1

2120₃ = 54 + 9 + 6 + 0

2120₃ = 69

Product means multiplication.

Change the numbers to base 10 and multiply and change it back to base seven.

124_seven = 1 x 7² + 2 x 7¹ + 4 x 7⁰

124_seven = 67

23_seven = 2 x 7¹ + 3 x 7⁰

23_seven = 17

124_seven x 23_seven = 67 x 17

124_seven x 23_seven = 1139

1139 base 10 to base 7

$$\begin{array}{cc}\mathrm{}& 1139\\ 7& \mathrm{162\; r\; 5}\\ 7& \mathrm{23\; r\; 1}\\ 7& \mathrm{3\; r\; 2}\\ 7& \mathrm{0\; r\; 3}\end{array}$$

You list the remainders from the bottom to the top.

1139_ten = 3215_seven

If you convert each of them to base 10, you should get the same value since they are equal.

2 x x¹ + 3 x x⁰ = 1 x 2³ + 1 x 2² + 1 x 2¹ + 1 x 2⁰

a⁰ = 1 (Every number raised to the power 0 is 1

a¹ = a (Every number to the power 1 is the same number)

2 x x + 3 x 1 = 1 x 8 + 1 x 4 + 1 x 2 + 1 x 1
2x + 3 = 8 + 4 + 2 + 1
2x = 8 + 4 + 2 + 1 - 3
2x = 12

Divide both sides by 2

x = $\frac{12}{2}$

x = 6

Method I

A number raised to the power 2 (square) means the number multiplied by itself.

Note: 2 in base two is 10.

Method II

Change the binary (base two) to base 10 and simplify and change the result back to base two.

11_two = 1 x 2¹ + 1 x 2⁰

11_two = 1 x 2 + 1 x 1

Note: any number raised to the power 0 is 1.

11_two = 2 + 1 = 3

(11_two)² = 3² = 3 x 3 = 9

Change the 9 base 10 to base 2.

9 base 10 to base 2

$$\begin{array}{cc}\mathrm{}& 9\\ 2& \mathrm{4\; r\; 1}\\ 2& \mathrm{2\; r\; 0}\\ 2& \mathrm{1\; r\; 0}\\ 2& \mathrm{0\; r\; 1}\end{array}$$

You list the remainders from the bottom to the top.

9_ten = 1001_two

Change each of the bases to base 10.

101_two = 1 x 2² + 0 x 2¹ + 1 x 2⁰

Note: any number raised to the power 0 is 1.

101_two = 4 + 0 + 1

101_two = 5

12_y = 1 x y¹ + 2 x y⁰

12_y = y + 2

23_five = 2 x 5¹ + 3 x 5⁰

23_five = 10 + 3

23_five = 13

Substitute the values of each of the bases.

5 + (y + 2) = 13

5 + y + 2 = 13

y + 7 = 13

y = 13 - 7

y = 6

Change both bases to base 10 to determine the value of y.

23_{y = 2 x y¹ + 3 x y⁰}

Any number raised to the power 0 is 1.

23_{y = 2 x y + 3 x 1}

23_{y = 2y + 3}

1111_two = 1 x 2³ + 1 x 2² + 1 x 2¹ + 1 x 2⁰

1111_two = 8 + 4 + 2 + 1

1111_two = 15

2y + 3 = 15

2y = 15 - 3

2y = 12

Divide both sides by 2

y = $\frac{12}{2}$ = 6

Calculate the value of 123_five in base 10 and change the base 10 to base six to get it's value in base six (M).

123_five = 1 x 5² + 2 x 5¹ + 3 x 5⁰

123_five = 25 + 10 + 3 = 38

38 base 10 to base 6

$$\begin{array}{cc}\mathrm{}& 38\\ 6& \mathrm{6\; r\; 2}\\ 6& \mathrm{1\; r\; 0}\\ 6& \mathrm{0\; r\; 1}\end{array}$$

You list the remainders from the bottom to the top.

38_ten = 102_six

M = 102

10101 + 1001 = 11110

***** = 111001 - 11110

***** = 11011

Note: In base 2, when you borrow, the value is two (2).