Square Root and Cube Root
Powers and Roots.—When a product consists of the same factor repeated any number of times it is called a power of that factor.
7 × 7 is the second power, or the square of 7.
7 × 7 × 7 is the third power, or the cube of 7.
A power of a number is generally expressed by writing the number only once, and placing after it, above the line, a small figure to show how many factors are to be taken. The small figure is called an index.
Thus, 72 = 49; 73 = 343; 74 = 2401.
A number is called the square root of its square.
Since 72 = 49, the square root of 49 is 7.
The “square root of 49” is written √49.
Again, a number is called the cube root of its cube. 73 = 343. Therefore, the cube root of 343 is 7.
The “cube root of 343” is written ∛343.
A perfect square is a number whose square root is a whole number. A perfect cube is a number whose cube root is a whole number.
Square Root.—If a number can be put into prime factors, its square root can be written down by inspection.
Example: Find the square root of 27225.
| Since 27225 | = | 32 | × | 52 | × | 112. | |
| ∴ √27225 | = | 3 | × | 5 | × | 11 | = 165 Ans. |
Rule for Digits.—We know that √1 = 1, and √100 = 10. Therefore, the square root of any number which lies between 1 and 100 lies between 1 and 10; i.e., if a number contains one or two digits, its square root consists of one digit.
Similarly, since √100 = 10 and √10000 = 100, the square root of a number between 100 and 10000 lies between 10 and 100. That is, if a number contains three or four digits, its square root consists of two digits.
Proceeding in this way, we obtain a general result—viz., the square of a number has either twice as many digits as the number, or one less than twice as many.
Hence, to ascertain the number of digits in the square root of a perfect square, mark off the [879] digits in pairs, beginning from the right. Each pair marked off gives a digit in the square root; and, if there is an odd digit remaining, that digit also gives a digit in the square root.
Examples: There are three digits in the square root of 546121, and four in the square root of 5774409.
For, marking off the digits from the right, we get in the first case 54,61,21, giving three digits in the square root, and in the second case 5,77,44,09, the odd digit giving the fourth in the square root.
The method of finding the square root of a given number depends on the form of the square of the sum of two numbers.
Explanation: The square root of 144 is 12. Let us see how we found it.
12 = 1 ten + 2 units.
122 is the same as (10 + 2)2.
Let us square (10 + 2), that is, multiply 10 + 2 by 10 + 2.
| 10 | + | 2 | ||
| 10 | + | 2 | ||
| 102 | + | (10 × 2) | ||
| + | (10 × 2) | + 22 | ||
| 102 | + | 2(10 × 2) | + 22 | |
| Then, 122 = | 102 | + | 2(10 × 2) | + 22 |
Rule.—The square of any number made up of tens and units is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units.
Another Explanation: Find the square root of 45369.
Solution:
| 4·53·69 | ) | 213 | |
| 4 | |||
| 41 | 53 | ||
| 41 | |||
| 423 | 1269 | ||
| 1269 | |||
(1) Point off the number into periods of two figures each, as before.
(2) The square root of the first period is 2. 2 × 2 = 4. Write the 2 in the root and subtract the 4 from 4. Bring down the next period, 53.
(3) 2 × 2 = 4. (Remember the 4 is to be used as a trial divisor, being 2 × the tens.)
4 is contained in 5 about 1 time. Place 1 in the root, also on the right of the 4 in the divisor. Multiply 41 by 1. Subtract and bring down the next period.
(4) 2 × 21 = 42. 42 is the trial divisor. 126 ÷ 42 = about 3 times. Place the 3 in the root also at the right of the 42 in the divisor. Multiply out.
Square root = 213.
Cube Root.—The cube root of a number is one of the three equal factors of that number.
Thus, 5 is the cube root of 125, because 5 × 5 × 5 = 125.
The radical sign with a figure 3 over it (∛ ) means that the cube root of the number following it is to be taken.
∛125 reads, “The cube root of 125.”
If we can find the prime factors of any perfect cube, we can write down its cube root by inspection.
Example: Find the cube root of 74088.
| 8 | 74088 |
| 9 | 9261 |
| 3 | 1029 |
| 7 | 343 |
| 7 | 49 |
| 7 |
| ∴ | 74088 | = | 8 × 9 × 3 × 7 × 7 × 7 |
| = | 23 × 33 × 73 | ||
| ∴ | ∛74088 | = | 2 × 3 × 7 |
| = | 42 Ans. | ||
Rule for Digits.—Since 13 = 1 and 103 = 1000, therefore the cube of a number which lies between 1 and 10 lies between 1 and 1000, i. e., the cube of a number of one digit contains either one, two or three digits.
Again, since 103 = 1000 and 1003 = 1000000, the cube of a number of two digits contains either four, five, or six digits.
Proceeding in this way, we see that the cube of a number contains three times, or one less or two less than three times, as many digits as the number.
Hence, to find the number of digits in the cube root of a given number, we mark off the digits in sets of three, beginning at the decimal point, and marking both to the right and to the left.
FIGURES REPRESENTING THE PROCESSES OF FINDING CUBE ROOT
Thus, 289383 will be pointed off into two periods—289·383—and we readily see there will be only 2 figures in the root.
The simplest method of finding the cube root of numbers whose prime factors are not known is analogous to the method of finding square root, being based upon the form of the cube of the sum of two numbers.
Explanation: The cube root of 1728 is 12. Let us see how we found it.
12 = 1 ten + 2 units
123= (10 + 2)3
(10 + 2)3 means 10 + 2 × 10 + 2 × 10 + 2
| 10 | + | 2 | |||||
| 10 | + | 2 | |||||
| 102 | + | (10 × 2) | |||||
| + | (10 × 2) | + | 22 | ||||
| 102 | + | 2(10 × 2) | + | 22 | |||
| 10 | + | 2 | |||||
| 103 | + | 2(102 × 2) | + | (10 × 22) | |||
| + | (102 × 2) | + | 2(10 × 22) | + | 23 | ||
| 103 | + | 3(102 × 2) | + | 3(10 × 22) | + | 23 | |
That is, the cube of any number made up of tens and units equals—
The cube of the tens + three times the product of the square of the tens by the units + three times the product of the tens by the square of the units + the cube of the units,
or
tens3 + 3(tens2 × units) + 3(tens × units2) + units3.
For graphic illustration the geometrical representation of the cube of units and tens in the drawings is helpful.
After the process is understood, this short method of writing the work may be used by the pupil:
Example: Find the cube root of.0163956, carrying the root to 3 decimal places.
Work:
| .016 | ·395·600 | ) | .254+ |
| 8 | |||
| 1200 | 8395 | ||
| 300 | |||
| 25 | |||
| 1525 | 7625 | ||
| 770600 | |||
| 187500 | |||
| 3000 | |||
| 16 | |||
| 190516 | 762064 | ||
| 8536 | |||