Convert each of the following repeating decimals to a/b form, where a, b are integers and b ≠ 0.

#2a ¯
0.4
#2c ¯
1.396
#2e ¯
-2.34

0.4 = 4/10 = 2/5.

1.396 = 1 396/1000 = 1396/1000 = 349/250

2.34 = 2 34/100 = 234/100 = 117/50

To convert repeating decimals to the a/b form, we need to follow a general procedure. Let's go step by step for each given decimal.

#2a) 0.4¯

Step 1: Identify the decimal as a repeating decimal.
In this case, there is a bar over the digit 4, indicating that it repeats indefinitely.

Step 2: Assign the variable x to the repeating part of the decimal.
Let x = 0.44¯

Step 3: Multiply both sides of the equation by a power of 10 that eliminates the repeating part.
Since there is only one digit repeating, we multiply by 10 to eliminate the repeating digit.
10x = 4.4¯

Step 4: Subtract the equation obtained in step 3 from the original equation obtained in step 2.
10x - x = 4.4¯ - 0.44¯
9x = 4

Step 5: Solve for x.
x = 4/9

So, 0.4¯ can be written as 4/9.

#2c) 1.396¯

Step 1: Identify the decimal as a repeating decimal.
In this case, there is a bar over the digits 396, indicating that it repeats indefinitely.

Step 2: Assign the variable x to the repeating part of the decimal.
Let x = 1.396¯

Step 3: Multiply both sides of the equation by a power of 10 that eliminates the repeating part.
Since there are three digits repeating, we multiply by 1000 to eliminate the repeating digits.
1000x = 1396.396¯

Step 4: Subtract the equation obtained in step 3 from the original equation obtained in step 2.
1000x - x = 1396.396¯ - 1.396¯
999x = 1395

Step 5: Solve for x.
x = 1395/999

So, 1.396¯ can be written as 1395/999.

#2e) -2.34¯

Step 1: Identify the decimal as a repeating decimal.
In this case, there is a bar over the digits 34, indicating that it repeats indefinitely.

Step 2: Assign the variable x to the repeating part of the decimal.
Let x = -2.3434¯

Step 3: Multiply both sides of the equation by a power of 10 that eliminates the repeating part.
Since there are two digits repeating, we multiply by 100 to eliminate the repeating digits.
100x = -234.3434¯

Step 4: Subtract the equation obtained in step 3 from the original equation obtained in step 2.
100x - x = -234.3434¯ - (-2.3434¯)
99x = -232.00

Step 5: Solve for x.
x = -232/99

So, -2.34¯ can be written as -232/99.