Square Root of 4900 — Value, Steps, and Why It Is 70
Square Root of 4900 — Value, Steps, and Why It Is 70
TL;DR
The square root of 4900 is exactly 70, because 70×70=4900. Since 4900 is a perfect square, its root is a whole, rational number, and this article shows three ways to reach 70 plus the mistakes that make students miss it.
What Is the Square Root of 4900?
The square root of 4900 is ( \sqrt{4900} = 70 ). It is exact, not an approximation, because 4900 is a perfect square: an integer multiplied by itself.
Quick Answer:
Result: ( \sqrt{4900} = 70 )
Notation: ( \sqrt{4900} = 70 ), or ( 4900^{1/2} = 70 )
Method shown: Prime factorization, grouping into perfect squares, long division
Rational or irrational: Rational (70 is a whole number)
Exact form: 70 (no decimal tail; nothing to approximate)
Every positive number technically has two square roots, one positive and one negative, so ( 70^2 = 4900 ) and ( (-70)^2 = 4900 ). The symbol ( \sqrt{\phantom{x}} ) means the principal (positive) root, so ( \sqrt{4900} = 70 ).
Quick Reference Table
The table sets 4900 among nearby perfect squares so the pattern is visible. Notice how the roots climb by 10 each time the base jumps to the next hundreds-square.
| Number | Square root | Exact or approximate |
|---|---|---|
| ( \sqrt{3600} ) | 60 | Exact (perfect square) |
| ( \sqrt{4225} ) | 65 | Exact (perfect square) |
| ( \sqrt{4761} ) | 69 | Exact (perfect square) |
| ( \sqrt{4900} ) | 70 | Exact (perfect square) |
| ( \sqrt{5041} ) | 71 | Exact (perfect square) |
| ( \sqrt{6400} ) | 80 | Exact (perfect square) |
| ( \sqrt{4899} ) | ≈69.99 | Approximate (not a perfect square) |
Where the Square Root of 4900 Shows Up
( \sqrt{4900} = 70 ) appears any time an area of 4900 square units must fold back into a side length: a square field of 4900 m² has sides of exactly 70 metres. It also shows up in the Pythagorean theorem whenever a squared distance works out to 4900, and in scaling problems where a quantity grows by a factor of 4900 in area but only 70 in length.
What Does "Square Root" Mean Here?
A square root of a number is a value that, multiplied by itself, gives that number. The square root undoes squaring.
A perfect square is any integer you get by multiplying an integer by itself, so 4900 is a perfect square because ( 70×70=4900 ). Perfect squares are exactly the numbers whose square roots are whole numbers, which is why ( \sqrt{4900} = 70 ) while ( \sqrt{4899} ) never terminates.
Is the square root of 4900 rational or irrational?
It is rational. A rational number can be written as a fraction of integers, and 70 is ( \frac{70}{1} ). Only the square roots of non-perfect squares, like √47 or √640, are irrational.
How To Compute the Square Root of 4900
Three methods land on the same answer. Pick the one that matches what you already know.
Method 1: Prime factorization
Break 4900 into primes.
( 4900 = 49 × 100 )
( 49 = 7 × 7 )
( 100 = 2 × 2 × 5 × 5 )
( 4900 = 2^2 × 5^2 × 7^2 )
Take one factor out of each pair.
( \sqrt{4900} = 2 × 5 × 7 = 70 )
Final answer: ( \sqrt{4900} = 70 )
Method 2: Grouping into known perfect squares
Split 4900 into a product of perfect squares you already recognize.
( 4900 = 49 × 100 )
( \sqrt{4900} = \sqrt{49} × \sqrt{100} )
( = 7 × 10 = 70 )
Final answer: ( \sqrt{4900} = 70 )
Method 3: Long division
Pair the digits from the right: ( \overline{49} \ \overline{00} ).
The largest square under 49 is 49, and ( 49=7 ), so the first digit is 7.
Bring down 00; the remainder after ( 49−49=0 ) leaves 00.
Double the 7 to get 14, and find a digit d with ( 14d × d ≤ 00), giving d=0.
The quotient reads 7070.
Final answer: ( \sqrt{4900} = 70 )
Common Mistakes With the Square Root of 4900
Mistake 1: Halving instead of rooting
Where it slips in: Rushing, a student reads ( \sqrt{4900} ) and divides by 2, writing 2450.
Don't do this: ( \sqrt{4900} ≠ 2450 )
The correct way: Ask what number times itself gives 4900. That is 70.
Mistake 2: Dropping a zero
Where it slips in: Students tack on one zero and answer 7 or 700 instead of 70.
Don't do this: Guess the zero count.
The correct way: Use ( 4900=49×100 ), so ( \sqrt{4900} = 7 × 10 = 70 ).
Mistake 3: Calling it irrational
Where it slips in: Assume every square root is a messy decimal after a run of irrational roots.
Don't do this: Just check if the number is a perfect square first.
The correct way: When it grabs paired primes, the root is a whole number.
Conclusion
- The square root of 4900 is exactly 70.
- 4900 is a perfect square, so its root is a rational whole number, not a decimal.
- The frequent errors are halving, mishandling the trailing zeros, and assuming the root is irrational.
Frequently Asked Questions
Is the square root of 4900 rational or irrational?
Rational. ( \sqrt{4900} = 70 ).
What is the square of 4900?
( 4900^2 = 24,010,000 ).
What is the square root of 490?
( \sqrt{490} \approx 22.14 ), an irrational number.
Why is 4900 a perfect square?
Because ( 4900=70×70 ) and its prime factorization has every prime in a pair.
What is the value of ( \sqrt{4900} × \sqrt{4900} )?
It is 4900.