How to Calculate the Square Root of 64: Simple Tricks for Students
The concept of square root of 64 is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Whether you are revising for board exams or want to master competitive maths, understanding how to find and use square roots like √64 is an important skill.
Understanding Square Root of 64
A square root of 64 refers to a value that, when multiplied by itself, gives 64. In mathematical terms, if \( x^2 = 64 \), then \( x = \sqrt{64} \). This concept is widely used in algebra, geometry, and number properties. Applications include calculating areas, solving quadratic equations, and simplifying mathematical expressions. Knowing the square root of perfect squares like 64 also helps students solve root and power-based questions quickly in school and competitive exams.
Formula Used in Square Root of 64
The standard formula is: \( \sqrt{64} = x \), where \( x \times x = 64 \).
Here’s a helpful table to understand square root of 64 and other perfect squares more clearly:
Square Roots of Perfect Squares Table
| Number | Square Root | Perfect Square? |
|---|---|---|
| 36 | 6 | Yes |
| 64 | 8 | Yes |
| 125 | ~11.18 | No |
| 81 | 9 | Yes |
| 144 | 12 | Yes |
This table shows how the square root of 64 fits among both smaller and larger perfect squares, making pattern recognition easy for revision and MCQ practice.
How to Calculate Square Root of 64 (Step By Step)
You can find the square root of 64 using different methods. Here are the most common, explained stepwise:
1. Prime Factorization Method:
a) Write 64 as a product of prime numbers: 64 = 2 × 2 × 2 × 2 × 2 × 2 = 26
b) Group into pairs of equal primes: (2 × 2) × (2 × 2) × (2 × 2)
c) Take one value from each pair: 2 × 2 × 2 = 8
So, \( \sqrt{64} = 8 \)
2. Repeated Subtraction Method:
Subtract consecutive odd numbers from 64 until you reach 0:
64 – 1 = 63
63 – 3 = 60
60 – 5 = 55
55 – 7 = 48
48 – 9 = 39
39 – 11 = 28
28 – 13 = 15
15 – 15 = 0
Number of steps = 8, so \( \sqrt{64} = 8 \)
3. Long Division Method:
a) Pair the digits (for 64, just one pair: 64).
b) Find the largest number whose square is ≤ 64 (8 × 8 = 64).
c) 8 gives a remainder of 0, so \( \sqrt{64} = 8 \)
You can find more about root calculation methods at How to find square root of a number and Square root long division method.
Worked Example – Solving a Problem
Let’s apply square root of 64 to a practical question:
Question: A class of 64 students must be seated in equal rows and columns (square). How many in each row/column?
1. Let the number of rows (and columns) be \( x \).
2. Since seating forms a square, \( x \times x = 64 \), so \( x = \sqrt{64} \).
3. \( \sqrt{64} = 8 \).
Answer: The class can be seated in 8 rows and 8 columns.
Practice Problems
- Is the square root of 36 less than square root of 64?
- Find the value of 3 × √64.
- What is the square root of 64 divided by 2?
- Which numbers between 1 and 100 have whole number square roots?
Common Mistakes to Avoid
- Confusing square root of 64 with the cube root of 64 (cube root of 64 = 4).
- Forgetting that every positive real number has both a positive and negative square root, but the principal square root is always positive.
- Writing only radical form and skipping simplification (√64 = 8, not just √64).
Real-World Applications
The square root of 64 is used in geometry (e.g., calculating the side length of a square with area 64 units²), measurement, construction, and in board and competitive exams. Understanding perfect squares helps in unit conversions, digital technology, and mathematical modeling. Vedantu helps students see how maths concepts like square roots are not just textbook ideas but tools used everywhere.
We explored the idea of square root of 64, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts and excel in examinations.
Quick Reference and Related Pages
- Square root of 4 – Compare calculation for smaller perfect squares.
- Cube root of 64 – Avoid confusion between square and cube roots.
- Square root of 576 – See patterns for higher perfect squares.
- Square root table – Get quick values for exam revision.
- Factors of 64 – Learn about factors and their use in root calculations.
- Squares and square roots – Deepen your understanding of related concepts.
- Value of root 2 – See a contrast between rational and irrational square roots.
- How to find square root of a number – Master all calculation methods.
- Table of 64 – Explore multiplication and squaring for this number.
FAQs on Square Root of 64 – Fast Calculation Methods and Key Concepts
1. What is the square root of 64?
The square root of 64 is 8. This means √64 = 8, because when 8 is multiplied by itself (8 × 8), it equals 64.
2. How do I calculate the square root of 64?
You can calculate the square root of 64 using several methods such as:
- Prime factorization: Express 64 as 2 × 2 × 2 × 2 × 2 × 2, group in pairs and multiply one number from each pair to get 8.
- Long division method: Dividing 64 stepwise to find its square root as 8.
- Repeated subtraction: Subtract successive odd numbers from 64 until reaching zero. The count of subtractions is the square root.
3. Is 64 a perfect square?
Yes, 64 is a perfect square because its square root is a whole number, specifically 8. A perfect square is a number obtained by squaring an integer.
4. What is the square root of 64 in simplified form?
The simplified form of the square root of 64 is just 8 because 64 is a perfect square. Expressed in radical form, it is √64, which simplifies directly to 8.
5. What is the value of √64 in decimal?
The value of √64 in decimal form is exactly 8.0, as 64 is a perfect square and its root is a whole number.
6. How do you solve 3√64?
To solve 3√64, multiply 3 by the square root of 64:
3 × √64 = 3 × 8 = 24. Always find the root first, then multiply.
7. Why do students confuse square root of 64 with cube root of 64?
The confusion arises because both involve roots of the number 64 but differ in operation:
- The square root (√64) asks for a number multiplied by itself to give 64, which is 8.
- The cube root (³√64) asks for a number multiplied by itself three times to give 64, which is 4.
Understanding the difference in powers clarifies this common misconception.
8. Why does the square root of 64 have only one positive value in exams?
In exams, the term square root usually refers to the principal (positive) square root. Although mathematically both positive and negative roots exist (±8 for 64), the positive root is used because it is the standard convention for clarity and simplicity.
9. Can square root of 64 be written as a fraction?
Yes, since √64 = 8 is an integer, it can also be expressed as the fraction 8/1. More generally, square roots can sometimes be written as simplified fractions when they result in rational numbers.
10. Why is √64 important for geometry problems?
√64 = 8 is important in geometry because:
- It helps calculate lengths of sides in squares, rectangles, and other shapes where area or dimensions relate to squares.
- It simplifies computations involving Pythagoras’ theorem where side lengths can be 64 or its multiples.
- Understanding perfect squares like 64 aids in solving many geometric problems quickly and accurately.
11. How can I quickly recognize perfect squares like 64 in multiple-choice questions?
To quickly identify perfect squares such as 64:
- Familiarize yourself with square numbers up to at least 15² (225), including 64 = 8².
- Notice that perfect squares end with digits 0, 1, 4, 5, 6, or 9 but never 2, 3, 7, or 8.
- Use shortcuts like memorizing squares of single-digit numbers for rapid recall under exam conditions.
