How to Find Square Root and Cube Root: Quick Techniques and Tips
Square Roots and Cube Roots Symbol
Square Root
The square root of a number x is that number which when multiplied by itself gives the number x itself. The number x is a perfect square.
For Example, 22 =4, or the square root of 4 is 2
32 =9, or the square root of 9 is 3
42 = 16, or the square root of 16 is 4
The symbol of the square root is \[\sqrt{}\]
Therefore, the square root of 4 is represented as \[\sqrt{4}\] = 2.
And the square root of 9 is represented as \[\sqrt{9}\] = 3 and so on.
Cube Root
The cube root of a number a is that number which when multiplied by itself three times gives the number ‘a’ itself.
For Example, 23 =8, or the cube root of 8 is 2
33 = 27, or the cube root of 27 is 3
43 = 64, or the cube root of 64 is 4
The symbol of the cube root is a 1/3 or ∛
Thus, the cube root of 8 is represented as \[\sqrt[3]{8}\] = 2 and that of 27 is represented as \[\sqrt[3]{27}\]= 3
and so on.
Properties of Square Root:
If the unit digit of a number is 2, 3, 7, and 8 then its square root is not a natural number.
If a number ends in an odd number of zeros, then its square root is not a natural number.
The square root of an even number is even and that of an odd number is odd.
Negative numbers have no squares root in a set of real numbers.
Properties of Cube Root
Cube root of all the odd numbers is an odd number. For example: \[\sqrt[3]{27}\]= 3, \[\sqrt[3]{125}\] = 5.
Cube root of all the even natural numbers is even. For example: \[\sqrt[3]{8}\] = 2,\[\sqrt[3]{64}\] = 4.
The cube root of a negative integer always results in negative.
Methods of Finding the Squares Roots and Cube Roots:
First, five number’s square root and cube root are easy to remember. But when we have to find square root and cube root of large numbers we can use the following methods to find it. They are
Prime Factorisation Method
Long Division method
This method will help you to find the square roots and cube root of a given number but if the square root and cube root of the first 10 numbers are memorized it will help you solve your problems more quickly. Here is the square root and cube root table format, which will help you to memorize these square roots and cube roots.
Square Root and Cube Root of 1 to 15
Square root list and cube root list of 1 to 15 will help you to solve the most time consuming long equations within no time. This square root and cube root table will be beneficial to you at every step.
Square Root and Cube Root Table
Make use of this square root list and cube root list and solve the below square root and cube root example.
Solved Examples
Square root and Cube root example
Example 1: Find the cube root of 2744
Solution :
By Prime Factorisation method
Step 1: First we take the prime factors of a given number
2744 = 2 x 7 x 2 x 2 x 7 x 7
Step 2: Form groups of three similar factors
= 2 x 2 x 2 x 7 x 7 x 7
Step 3: Take out one factor from each group and multiply.
= 23 x 73
= 143
Therefore, \[\sqrt[3]{2744}\]= 14
Example 2: Find the cube root of 1728 by long division method
Solution:
Now,
\[\sqrt[3]{1728}\]= \[\sqrt[3]{2X2X2X2X2X2X3X3}\]
= 2 x 2 x 3
= 12
Quiz
Some more square root and cube root examples to solve.
Estimate the square root of 666
Find the value of \[\sqrt{256}\]
Using prime factorization, find the value of \[\sqrt[3]{1331}\]
Using long division method, find the value of \[\sqrt[3]{729}\]
FAQs on Square Root and Cube Root Made Simple
1. What is the fundamental difference between a square root and a cube root?
The fundamental difference lies in the operation they reverse. A square root reverses squaring a number, while a cube root reverses cubing a number. Here are the key distinctions:
- Definition: The square root of a number 'x' is a value 'y' such that y × y = x. The cube root of 'x' is a value 'z' such that z × z × z = x.
- Inverse Operation: Square root is the inverse of squaring (raising to the power of 2). Cube root is the inverse of cubing (raising to the power of 3).
- Notation: The square root is denoted by the radical symbol (√), e.g., √25 = 5. The cube root is denoted by a radical with a small 3 (∛), e.g., ∛27 = 3.
- Geometric Meaning: The square root of an area gives the side length of a square. The cube root of a volume gives the edge length of a cube.
2. What are perfect squares and perfect cubes? Give some examples.
A perfect square is a whole number that is the result of squaring another whole number. A perfect cube is a whole number that is the result of cubing another whole number.
- Examples of Perfect Squares: 9 (since 3² = 9), 16 (since 4² = 16), and 144 (since 12² = 144).
- Examples of Perfect Cubes: 8 (since 2³ = 8), 64 (since 4³ = 64), and 1000 (since 10³ = 1000).
A number like 10 is neither a perfect square nor a perfect cube because it cannot be obtained by squaring or cubing any whole number.
3. How are square roots of perfect squares calculated using the prime factorisation method?
The prime factorisation method is a systematic way to find the square root of a perfect square. Here are the steps with an example for finding the square root of 324:
- Step 1: Find the prime factors of the number. For 324, the prime factors are 2 × 2 × 3 × 3 × 3 × 3.
- Step 2: Group the identical factors into pairs. Here, we get (2 × 2) × (3 × 3) × (3 × 3).
- Step 3: Take one factor from each pair. From the pairs, we take one 2, one 3, and another 3.
- Step 4: Multiply these selected factors together. So, 2 × 3 × 3 = 18.
Therefore, the square root of 324 is 18.
4. What is the method for finding the cube root of a perfect cube?
To find the cube root of a perfect cube using prime factorisation, you group the factors in triplets. Let's find the cube root of 216:
- Step 1: Determine the prime factors of the number. For 216, the prime factors are 2 × 2 × 2 × 3 × 3 × 3.
- Step 2: Group the identical factors into triplets (groups of three). This gives us (2 × 2 × 2) × (3 × 3 × 3).
- Step 3: Take one factor from each triplet. We take one 2 and one 3.
- Step 4: Multiply these factors to get the final answer. So, 2 × 3 = 6.
Thus, the cube root of 216 is 6.
5. Why can't a negative number have a real square root, but it can have a real cube root?
This is because of the rules of multiplication with negative numbers.
- For Square Roots: To find the square root of a number, you need another number that, when multiplied by itself, equals the original number. Multiplying a positive number by itself results in a positive (e.g., 5 × 5 = 25). Multiplying a negative number by itself also results in a positive (e.g., -5 × -5 = 25). It is impossible to get a negative result, so a negative number like -25 has no real square root.
- For Cube Roots: To find a cube root, a number is multiplied by itself three times. A negative number multiplied three times results in a negative number. For example, to find ∛-8, the answer is -2 because (-2) × (-2) × (-2) = 4 × (-2) = -8.
6. How can you estimate the square root of a non-perfect square like 60?
You can estimate the square root of a non-perfect square by finding the two closest perfect squares it lies between.
- Step 1: Identify the perfect squares just below and just above 60. The closest perfect square below 60 is 49 (which is 7²), and the closest perfect square above 60 is 64 (which is 8²).
- Step 2: This tells us that the square root of 60 must be a number between 7 and 8.
- Step 3: To get a better estimate, see which perfect square 60 is closer to. The distance from 60 to 64 is 4, while the distance from 60 to 49 is 11. Since 60 is closer to 64, its square root will be closer to 8 than to 7. We can estimate it to be around 7.7 or 7.8.
7. In what real-world scenarios would you need to calculate a square root or a cube root?
Square roots and cube roots have practical applications in various fields:
- Square Root Applications: They are crucial in geometry and physics. For example, using the Pythagorean theorem to find the length of a diagonal, calculating distances, or finding the side length of a square piece of land if you only know its total area.
- Cube Root Applications: These are most common in scenarios involving three-dimensional space. For instance, if you know the volume of a cubic box or shipping container, you can find the length of one of its edges by calculating the cube root. This is useful in design, packaging, and engineering.
8. How does knowing the unit digit of a number help in quickly identifying its possible square root or cube root?
The unit digit of a number provides a useful clue about the unit digit of its root. This is a great shortcut for checking answers or making estimations.
- For Square Roots: A perfect square ending in 1 must have a square root ending in 1 or 9. A number ending in 6 must have a root ending in 4 or 6. Numbers ending in 2, 3, 7, or 8 can never be perfect squares.
- For Cube Roots: The pattern is even simpler. Each unit digit corresponds to a unique unit digit in its cube root. For example, a perfect cube ending in 8 will always have a cube root ending in 2 (e.g., ∛8 = 2, ∛1728 = 12). Similarly, a number ending in 7 must have a cube root ending in 3 (e.g., ∛27 = 3).
