Square Root & Cube Root
Step 1: First of all group the number in pairs of 2 starting from the right.
Step 2: To get the ten’s place digit, Find the nearest square (equivalent or greater than or less than) to the first grouped pair from left and put the square root of the square.
Step 3: To get the unit’s place digit of the square root
Remember the following
| If number ends in | Unit’s place digit of the square root |
| 1 | 1 or 9(10-1) |
| 4 | 2 or 8(10-2) |
| 9 | 3 or 7(10-3) |
| 6 | 4or 6(10-4) |
| 5 | 5 |
| 0 | 0 |
Lets see the logic behind this for a better understanding
We know,
12=1
22=4
32=9
42=16
52=25
62=36
72=49
82=64
92=81
102=100
Now, observe the unit’s place digit of all the squares.
Do you find anything common?
We notice that,
Unit’s place digit of both 12 and 92 is 1.
Unit’s place digit of both 22 and 82 is 4
Unit’s place digit of both 32 and 72 is 9
Unit’s place digit of both 42 and 62 is 6.
Step 4: Multiply the ten’s place digit (found in step 1) with its consecutive number and compare the result obtained with the first pair of the original number from left.
Remember,
If first pair of the original number > Result obtained on multiplication then select the greater number out of the two numbers as the unit’s place digit of the square root.
If firstpair of the original number < the result obtained on multiplication,then select the lesser number out of the two numbers as the unit’s place digit of the square root.
Let us consider an example to get a better understanding of the method
Example 1: √784=?
Step 1: We start by grouping the numbers in pairs of two from right as follows
7 84
Step 2: To get the ten’s place digit,
We find that nearest square to first group (7) is 4 and √4=2
Therefore ten’s place digit=2
Step 3: To get the unit’s place digit,
We notice that the number ends with 4, So the unit’s place digit of the square root should be either 2 or 8(Refer table).
Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(2) and its consecutive number(3) we get,
2×3=6
ten’s place digit of original number > Multiplication result
7>6
So we need to select the greater number (8) as the unit’s place digit of the square root.
Unit’s place digit =8
Ans:√784=28
Cube roots of perfect cubes
It may take two-three minutes to find out cube root of a perfect cube by using conventional method. However we can find out cube roots of perfect cubes very fast, say in one-two seconds using Vedic Mathematics.
We need to remember some interesting properties of numbers to do these quick mental calculations which are given below.
Points to remember for speedy calculation of cube roots
- To calculate cube root of any perfect cube quickly, we need to remember the cubes of 1 to 10 which is given below.
| 13 | = | 1 |
| 23 | = | 8 |
| 33 | = | 27 |
| 43 | = | 64 |
| 53 | = | 125 |
| 63 | = | 216 |
| 73 | = | 343 |
| 83 | = | 512 |
| 93 | = | 729 |
| 103 | = | 1000 |
- From the above cubes of 1 to 10, we need to remember an interesting property.
| 13 = 1 | => | If last digit of the perfect cube = 1, last digit of the cube root = 1 |
| 23 = 8 | => | If last digit of the perfect cube = 8, last digit of the cube root = 2 |
| 33 = 27 | => | If last digit of the perfect cube = 7, last digit of the cube root = 3 |
| 43 = 64 | => | If last digit of the perfect cube = 4, last digit of the cube root = 4 |
| 53 = 125 | => | If last digit of the perfect cube =5, last digit of the cube root = 5 |
| 63 = 216 | => | If last digit of the perfect cube = 6, last digit of the cube root = 6 |
| 73 = 343 | => | If last digit of the perfect cube = 3, last digit of the cube root = 7 |
| 83 = 512 | => | If last digit of the perfect cube = 2, last digit of the cube root = 8 |
| 93 = 729 | => | If last digit of the perfect cube = 9, last digit of the cube root = 9 |
| 103 = 1000 | => | If last digit of the perfect cube = 0, last digit of the cube root = 0 |
It’s very easy to remember the relations given above because
| 1 | -> | 1 | (Same numbers) |
| 8 | -> | 2 | (10’s complement of 8 is 2 and 8+2 = 10) |
| 7 | -> | 3 | (10’s complement of 7 is 3 and 7+3 = 10) |
| 4 | -> | 4 | (Same numbers) |
| 5 | -> | 5 | (Same numbers) |
| 6 | -> | 6 | (Same numbers) |
| 3 | -> | 7 | (10’s complement of 3 is 7 and 3+7 = 10) |
| 2 | -> | 8 | (10’s complement of 2 is 8 and 2+8 = 10) |
| 9 | -> | 9 | (Same numbers) |
| 0 | -> | 0 | (Same numbers) |
Also see
8 -> 2 and 2 -> 8
7 -> 3 and 3-> 7
Questions
Level-I
| 1. | The cube root of .000216 is: | |||||||
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2. |
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| 3. | The least perfect square, which is divisible by each of 21, 36 and 66 is: | |||||||
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| 4. | 1.5625 = ? | |||||||
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| 5. | If 35 + 125 = 17.88, then what will be the value of 80 + 65 ? | |||||||
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6. |
If a = 0.1039, then the value of 4a2 – 4a + 1 + 3a is: |
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| 8. | A group of students decided to collect as many paise from each member of group as is the number of members. If the total collection amounts to Rs. 59.29, the number of the member is the group is: | |||||||
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| 9. | The square root of (7 + 35) (7 – 35) is | |||||||
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10. |
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Level-II
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| 12. | 0.0169 x ? = 1.3 | |||||||
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| 14. | How many two-digit numbers satisfy this property.: The last digit (unit’s digit) of the square of the two-digit number is 8 ? | |||||||
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| 15. | The square root of 64009 is: | |||||||
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| 16. √29929 = ? | ||||||||
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| 17. √106.09 = ? | ||
| A. | 10.6 | |
| B. | 10.5 | |
| C. | 10.3 | |
| D. | 10.2 | |
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18. ?/√196 = 5 |
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| A. | 76 | |
| B. | 72 | |
| C. | 70 | |
| D. | 75 | |
Answers
Level-I
Answer:1 Option B
Explanation:
| (.000216)1/3 | = | 216 | 1/3 | ||
| 106 |
| = | 6 x 6 x 6 | 1/3 | ||
| 102 x 102 x 102 |
| = | 6 |
| 102 |
| = | 6 |
| 100 |
= 0.06
Answer:2 Option A
Explanation:
| Let | x | = | 162 |
| 128 | x |
Then x2 = 128 x 162
= 64 x 2 x 18 x 9
= 82 x 62 x 32
= 8 x 6 x 3
= 144.
x = 144 = 12.
Answer:3 Option A
Explanation:
L.C.M. of 21, 36, 66 = 2772.
Now, 2772 = 2 x 2 x 3 x 3 x 7 x 11
To make it a perfect square, it must be multiplied by 7 x 11.
So, required number = 22 x 32 x 72 x 112 = 213444
Answer:4 Option B
Explanation:
1|1.5625( 1.25
|1
|——-
22| 56
| 44
|——-
245| 1225
| 1225
|——-
| X
|——-
1.5625 = 1.25.
Answer:5 Option D
Explanation:
35 + 125 = 17.88
35 + 25 x 5 = 17.88
35 + 55 = 17.88
85 = 17.88
5 = 2.235
80 + 65 = 16 x 5 + 65
= 45 + 65
= 105 = (10 x 2.235) = 22.35
Answer:6 Option C
Explanation:
4a2 – 4a + 1 + 3a = (1)2 + (2a)2 – 2 x 1 x 2a + 3a
= (1 – 2a)2 + 3a
= (1 – 2a) + 3a
= (1 + a)
= (1 + 0.1039)
= 1.1039
Answer:7 Option C
Explanation:
| x = | (3 + 1) | x | (3 + 1) | = | (3 + 1)2 | = | 3 + 1 + 23 | = 2 + 3. |
| (3 – 1) | (3 + 1) | (3 – 1) | 2 |
| y = | (3 – 1) | x | (3 – 1) | = | (3 – 1)2 | = | 3 + 1 – 23 | = 2 – 3. |
| (3 + 1) | (3 – 1) | (3 – 1) | 2 |
x2 + y2 = (2 + 3)2 + (2 – 3)2
= 2(4 + 3)
= 14
Answer:8 Option C
Explanation:
Money collected = (59.29 x 100) paise = 5929 paise.
Number of members = 5929 = 77
Answer:9 Option B
Explanation:
| (7 + 35)(7 – 35) | = | (7)2 – (35)2 | = 49 – 45 = 4 = 2 |
Answer:10 Option B
Explanation:
| 5 | – | 10 | + 125 | = | (5)2 – 20 + 25 x 55 |
| 2 | 5 | 25 |
| = | 5 – 20 + 50 |
| 25 |
| = | 35 | x | 5 |
| 25 | 5 |
| = | 355 |
| 10 |
| = | 7 x 2.236 |
| 2 |
| = 7 x 1.118 | |
| = 7.826 | |
Level-II
Answer:11 Option A
Explanation:
| Given Expression = | 25 | x | 14 | x | 11 | = 5. |
| 11 | 5 | 14 |
Answer:12 Option B
Explanation:
Let 0.0169 x x = 1.3.
Then, 0.0169x = (1.3)2 = 1.69
| x = | 1.69 | = 100 |
| 0.0169 |
Answer:13 Option C
Explanation:
| 3 – | 1 | 2 | = (3)2 + | 1 | 2 | – 2 x 3 x | 1 | ||||
| 3 | 3 | 3 |
| = 3 + | 1 | – 2 |
| 3 |
| = 1 + | 1 |
| 3 |
| = | 4 |
| 3 |
Answer:14 Option D
Explanation:
A number ending in 8 can never be a perfect square.
Answer:15 Option A
Explanation:
2 |64009( 253 |4 |———-45 |240 |225 |———-503| 1509 | 1509 |———- | X |———-
64009 = 253.
Answer:16 Option A
Explanation:
√29929 = So, √29929 = 173
Answer:17 Option C
Answer:18 Option C,
Square roots and cube roots are two types of roots. A square root of a number is a number that, when multiplied by itself, equals the original number. A cube root of a number is a number that, when multiplied by itself three times, equals the original number.
- Definition of square root and cube root
A square root of a number $x$ is a number $y$ such that $y^2 = x$. In other words, a square root of $x$ is a number that, when multiplied by itself, equals $x$.
A cube root of a number $x$ is a number $y$ such that $y^3 = x$. In other words, a cube root of $x$ is a number that, when multiplied by itself three times, equals $x$.
- Properties of square roots and cube roots
Some properties of square roots and cube roots are as follows:
- The square root of a non-negative number is always non-negative.
- The square root of a perfect square is an integer.
- The cube root of a non-negative number is always non-negative.
- The cube root of a perfect cube is an integer.
- The square root of a number is unique.
-
The cube root of a number is not always unique. For example, the cube roots of 8 are 2 and -2.
-
How to find square roots and cube roots
There are several ways to find square roots and cube roots. One way is to use a calculator. Another way is to use a table of square roots and cube roots.
To find the square root of a number using a calculator, enter the number and press the square root button. For example, to find the square root of 9, enter 9 and press the square root button. The calculator will display the answer, which is 3.
To find the cube root of a number using a calculator, enter the number and press the cube root button. For example, to find the cube root of 27, enter 27 and press the cube root button. The calculator will display the answer, which is 3.
To find the square root of a number using a table of square roots, look up the number in the table. The table will show you the square root of the number.
To find the cube root of a number using a table of cube roots, look up the number in the table. The table will show you the cube root of the number.
- Applications of square roots and cube roots
Square roots and cube roots are used in many different areas of mathematics, science, and engineering. Some applications of square roots and cube roots are as follows:
- In geometry, square roots are used to find the length of the diagonal of a square or rectangle.
- In physics, square roots are used to find the speed of a moving object.
- In chemistry, square roots are used to find the concentration of a solution.
-
In engineering, square roots are used to find the Stress in a beam.
-
Square root and cube root tables
Square root and cube root tables are tables that list the square roots and cube roots of numbers. Square root and cube root tables can be used to find the square root or cube root of a number without using a calculator.
- Square root and cube root calculators
Square root and cube root calculators are calculators that can be used to find the square root or cube root of a number. Square root and cube root calculators are available online and on many handheld calculators.
What is a function?
A function is a rule that takes an input and gives an output. For example, the function $f(x) = x^2$ takes an input $x$ and gives an output $x^2$.
What is a graph?
A graph is a visual representation of a function. It is made up of points that are connected by lines. The points represent the input values, and the lines represent the output values.
What is the domain of a function?
The domain of a function is the set of all possible input values. For example, the domain of the function $f(x) = x^2$ is all real numbers.
What is the range of a function?
The range of a function is the set of all possible output values. For example, the range of the function $f(x) = x^2$ is all non-negative real numbers.
What is the inverse of a function?
The inverse of a function is a function that undoes the original function. For example, the inverse of the function $f(x) = x^2$ is the function $g(x) = \sqrt{x}$.
What is a derivative?
The derivative of a function is a measure of how much the function changes as its input changes. For example, the derivative of the function $f(x) = x^2$ is $2x$.
What is an integral?
The integral of a function is a measure of the area under the curve of the function. For example, the integral of the function $f(x) = x^2$ from $0$ to $1$ is $\frac{1}{3}$.
What is a limit?
A limit is the value that a function approaches as its input approaches a certain value. For example, the limit of the function $f(x) = x^2$ as $x$ approaches $2$ is $4$.
What is a series?
A series is a sum of a number of terms. For example, the series $1 + 2 + 3 + \dots$ is called the harmonic series.
What is a sequence?
A sequence is a list of numbers that are ordered by their position in the list. For example, the sequence $1, 2, 3, \dots$ is called the natural numbers.
What is a Probability?
A probability is a measure of how likely an event is to occur. For example, the probability of rolling a six on a six-sided die is $\frac{1}{6}$.
What is a statistics?
Statistics is the study of the collection, organization, analysis, interpretation, presentation, and Communication of data.
What is a linear equation?
A linear equation is an equation of the form $ax + by = c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.
What is a quadratic equation?
A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.
What is a cubic equation?
A cubic equation is an equation of the form $ax^3 + bx^2 + cx + d = 0$, where $a$, $b$, $c$, and $d$ are real numbers and $a \neq 0$.
What is a system of equations?
A system of equations is a set of two or more equations that are related to each other. For example, the system of equations $x + y = 2$ and $x – y = 1$ has the solution $(1, 1)$.
What is a matrix?
A matrix is a rectangular array of numbers. For example, the matrix $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ has two rows and two columns.
What is a vector?
A vector is a quantity that has both magnitude and direction. For example, the vector $(1, 2)$ has a magnitude of $\sqrt{5}$ and a direction of $30^\circ$ counterclockwise from the positive $x$-axis.
What is a linear transformation?
A linear transformation is a function that maps vectors to vectors in a linear way. For example, the
Sure, here are some multiple choice questions about numbers and operations, without mentioning the topic of square roots and cube roots:
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What is the product of 2 and 3?
(A) 6
(B) 4
(C) 12
(D) 8 -
What is the sum of 1 and 2?
(A) 3
(B) 4
(C) 5
(D) 6 -
What is the difference of 5 and 2?
(A) 3
(B) 4
(C) 5
(D) 6 -
What is the quotient of 6 and 2?
(A) 3
(B) 4
(C) 5
(D) 6 -
What is the remainder when 7 is divided by 3?
(A) 1
(B) 2
(C) 3
(D) 4 -
What is the next number in the sequence 1, 2, 4, 8, 16?
(A) 32
(B) 64
(C) 128
(D) 256 -
What is the next number in the sequence 2, 3, 5, 7, 11?
(A) 13
(B) 17
(C) 19
(D) 23 -
What is the next number in the sequence 1, 4, 9, 16, 25?
(A) 36
(B) 49
(C) 64
(D) 81 -
What is the next number in the sequence 2, 6, 12, 20, 30?
(A) 42
(B) 52
(C) 66
(D) 78 -
What is the next number in the sequence 3, 6, 12, 21, 33?
(A) 46
(B) 55
(C) 66
(D) 78
I hope these questions were challenging and fun!