How to Estimate Square Roots and Cube Roots with Examples and Tricks
Square Roots
When two numbers are multiplied with each other, they form the square of the number. It is named square because if you feed the dimension in one side of the square, then the area of the corresponding square will give you the square of a number. When we break down these squares to their primitive numbers, then the resulting number is called a square root of the base number.
The traditional method of finding square roots is by prime factorisation the number and then pairing the same divisors into a single group and considering them as one; then they are all multiplied to give us the square root of the base number. Some square root examples:
\[\sqrt[2]{36}\] = 6
\[\sqrt[2]{81}\] = 9
Cube Roots
When three numbers are multiplied with itself, then the resulting figure is termed as the cube of the number. It is derived from the fact that, if the sides of a cube are taken as the number respectively then the resulting output is the volume of the cube. When this cube of a number is broken down to its primitive form, then the numbers so obtained are known as the cube root of the base number. The traditional method to find the cube root of a number is by prime factorising the cube number and then arranging the divisor into a group of three same numbers and then assuming them as a single entity. After that, all the individual pieces are multiplied, which gives us the output as a cube root. Some cube root examples:
\[\sqrt[3]{1728}\] = 12
\[\sqrt[3]{343}\] = 7
Estimating Square Root
The process of estimating square roots is quite easy; let’s approach it step by step. The first step to determine the root of a number is to arrange it into doublets from the right. After that, the numbers of doublets determine the number of digits in the root. After that, the nearest square is considered and from both directions, i.e. the largest perfect square near it and the smallest square after it. Then the square is estimated accordingly
Let’s understand it with an example.
For the estimation, we take the number 326.
Now we choose dual numbers from the right side, i.e. 3 26 since it has two bars; hence the root will have two numbers.
Now, we will estimate the nearest square root. We know that the square of the number 18 is 324 and the square of the number 19 is 361. Hence the number lies between 18 and 19.
If we observe carefully then the square of 18 is nearer to the number than the square of number 19; hence we can estimate the square root as 18.
Estimating Cube Root
For estimation of cube root, we need to follow the similar steps to that of estimating the square, but now instead of taking two numbers at a time, we will take three and then proceed by comparing the nearest cubes of the number.
Let’s try it with an example.
For the estimation process, we take the number 13824.
Now we break it into groups of three 13 824.
The last digit of the first triplet gives us the number 4, which means that the one’s place in our cube root is 4 itself because only the cube of 4 ends with 4 since the number is a two-digit number because of two bars we need to figure out the tens place next.
The number 13 is closest to the cube of 2 that is 8, and the next number closest is 3, which gives us 27 since we consider the lowest limit; hence the tens digit is 2.
Therefore we get the cube root as 3.
FAQs on Estimating Square Root and Cube Root Step by Step
1. What is the square root of a number?
The square root of a number is a value that, when multiplied by itself, gives the original number. In symbols, if x² = a, then x = √a.
- Example: Since 5 × 5 = 25, the square root of 25 is √25 = 5.
- Every positive number has two square roots: a positive and a negative one (e.g., ±5).
- The symbol for square root is √.
2. What is the cube root of a number?
The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In symbols, if x³ = a, then x = ∛a.
- Example: Since 3 × 3 × 3 = 27, the cube root of 27 is ∛27 = 3.
- Unlike square roots, cube roots of negative numbers are negative (e.g., ∛(−8) = −2).
3. How do you estimate the square root of a number that is not a perfect square?
To estimate the square root of a non-perfect square, find the two perfect squares it lies between and approximate its value.
- Step 1: Identify nearby perfect squares.
- Step 2: Compare distances to estimate the decimal value.
- Example: To estimate √20:
- Since 4² = 16 and 5² = 25, √20 lies between 4 and 5.
- 20 is closer to 16 than 25, so √20 ≈ 4.5 (more accurately 4.47).
4. How do you estimate the cube root of a number?
To estimate the cube root, locate the two nearest perfect cubes and determine where the number lies between them.
- Step 1: Identify nearby perfect cubes.
- Step 2: Compare distances for approximation.
- Example: To estimate ∛50:
- Since 3³ = 27 and 4³ = 64, ∛50 lies between 3 and 4.
- 50 is closer to 64, so ∛50 ≈ 3.7 (more accurately 3.68).
5. What is the difference between a square root and a cube root?
The key difference is that a square root involves multiplying a number by itself twice, while a cube root involves multiplying it three times.
- Square root: x² = a, so x = √a.
- Cube root: x³ = a, so x = ∛a.
- Square roots of positive numbers have two values (±).
- Cube roots have only one real value, even for negative numbers.
6. How do you find the square root using the long division method?
The long division method finds the square root by grouping digits in pairs and systematically dividing to get each digit of the root.
- Step 1: Pair digits from right to left.
- Step 2: Find the largest square less than or equal to the first pair.
- Step 3: Subtract and bring down the next pair.
- Step 4: Double the current root and determine the next digit.
- Example: Using this method, √144 = 12.
7. Can you give an example of estimating √45 step by step?
To estimate √45, compare it with nearby perfect squares and refine the value.
- 6² = 36 and 7² = 49.
- 45 lies between 36 and 49, so √45 is between 6 and 7.
- 45 is closer to 49, so estimate around 6.7.
- More accurate value: √45 ≈ 6.71.
8. What are perfect squares and perfect cubes?
A perfect square is a number obtained by squaring an integer, and a perfect cube is obtained by cubing an integer.
- Perfect squares: 1, 4, 9, 16, 25 (1², 2², 3², 4², 5²).
- Perfect cubes: 1, 8, 27, 64, 125 (1³, 2³, 3³, 4³, 5³).
- They have whole number square roots or cube roots.
9. Why do square roots of negative numbers not exist in real numbers?
The square root of a negative number is not a real number because no real number multiplied by itself gives a negative result.
- Positive × Positive = Positive.
- Negative × Negative = Positive.
- Therefore, √(−9) is not real.
- In complex numbers, √(−9) = 3i, where i² = −1.
10. What are some common mistakes when estimating square roots and cube roots?
Common mistakes when estimating square roots and cube roots include choosing incorrect nearby perfect powers and ignoring decimal refinement.
- Confusing perfect squares with perfect cubes.
- Not checking which value the number is closer to.
- Forgetting that √a has two values (±).
- Incorrectly estimating cube roots of negative numbers.
