When we refer to number less than one in this section, we mean in the range 0<n<1, not negative values.

Objective:

To find squares and square roots of numbers outside the normal range.

General:

• Finding squares and square roots of numbers less than 1 and greater than 100 is done using the exact same process of the slide rule.  However, we will need to rewrite the number in an appropriate first.

Method for Squares:

• Write the number in scientific notation.
• Determine the square of the decimal part as normal.
• Multiply the exponent of the power of ten by two.
• Multiply the result from the rule by the new power of ten.

Method for Square Roots:

• Write the number so that a power of ten divisible by 2 is the factored out without the decimal part not going less than 1.
• Determine the square root as normal.
• Divide the index of the power of ten by two.
• Multiply the result from the rule by the new power of ten.

 

Example 1:  Calculate 114²

Writing 114 in scientific notation gives 1.14 x 10².  Now find 1.14² as usual.

• Move the cursor to D1.14
• Read the result A1.3

• Multiply the exponent by 2 giving 4.
• Multiply the result by 10⁴
• Answer: 1.3 x 10⁴ = 13000

 

Example 2:  Calculate 4750²

Write 4750 in scientific notation giving 4.75 x 10³.  Now find 4.75² as usual.

• Move the cursor to D4.75
• Read the result A22.5

• Multiply the exponent by 2 giving 6.
• Multiply the result by 10⁶
• Answer 22.5 x 10⁶ = 22,500,000

 

Example 3:  Calculate 0.00255²

Write 0.00255 in scientific notation giving 2.55 x 10 ^-3.  Now find 2.55² as usual.

• Move the cursor to D2.55
• Read the result A6.5

• Multiply the exponent by 2 giving -6.
• Multiply the result by 10^-6
• Answer 6.5 x 10^{-6,/ = 0.0000065}

 

Example 4:  Calculate √235.

The power of ten with exponent divisible by 2 that can be factored out is 10², so we rewrite 235 as 2.35 x 10².

• Move the cursor to A2.35 (first decade)
• Read the result D1.53

• Divide the exponent by 2 giving 1.
• Multiply the result from the rule by 10¹.
• Answer 1.53 x 10¹ = 15.3

 

Example 5:  Calculate √455000.

The power of ten with exponent divisible by 2 that can be factored out is 10⁴, so we rewrite 455000 as 45.5 x 10⁴.  Now find √45.5 as normal.

• Move the cursor to A45.5 (second decade)
• Read the result D6.75

• Divide the exponent by 2 giving 2.
• Multiply the result by 10².
• Answer: 6.75 x 10² = 675

 

Example 6:  Calculate √0.000039.

The power of ten with exponent divisible by 2 that can be factored out is 10^-6, so we rewrite 0.000039 as 39 x 10^-6.  Now find √39 as normal.

• Move the cursor to A39 (second decade)
• Read the result D6.25

• Divide the exponent by 2 giving -3.
• Multiply the result by 10^-3.
• Answer: 6.25 x 10^-3 = 0.00625

 

Practice Questions:  Evaluate the following:

1. 225² (Ans: 50600)
2. 460² (Ans: 212000)
3. 0.0055² (Ans: 0.0000303)
4. √890 (Ans: 29.8)
5. √63,500 (Ans: 252)
6. √0.0000635 (Ans: 0.00797)