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, $60^2$) |
$\sqrt{4225}$ | $65$ | Exact (perfect square, $65^2$) |
$\sqrt{4761}$ | $69$ | Exact (perfect square, $69^2$) |
$\sqrt{4900}$ | $70$ | Exact (perfect square, $70^2$) |
$\sqrt{5041}$ | $71$ | Exact (perfect square, $71^2$) |
$\sqrt{6400}$ | $80$ | Exact (perfect square, $80^2$) |
$\sqrt{4899}$ | $\approx 69.99$ | 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 $4900 \text{ m}^2$ 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 $70 \times 70 = 4900$. Perfect squares are exactly the numbers whose square roots are whole numbers, which is why $\sqrt{4900}$ closes neatly to 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.
$4900 = 49 \times 100$
$49 = 7 \times 7$
$100 = 2 \times 2 \times 5 \times 5$
$4900 = 2^2 \times 5^2 \times 7^2$
Take one factor out of each pair.
$\sqrt{4900} = 2 \times 5 \times 7 = 70$
Final answer: $\sqrt{4900} = 70$
Method 2: Grouping into known perfect squares
Split 4900 into a product of perfect squares you already recognise.
$4900 = 49 \times 100$
$\sqrt{4900} = \sqrt{49} \times \sqrt{100}$
$\sqrt{4900} = 7 \times 10$
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 $\sqrt{49} = 7$, 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 $14d \times d \le 0$, giving $d = 0$.
The quotient reads $70$.
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: $\sqrt{4900} = 4900 \div 2 = 2450$. Squaring undoes a square root, and $2450^2$ is nowhere near 4900.
The correct way: Ask what number times itself gives 4900. That is 70, since $70 \times 70 = 4900$.
Mistake 2: Dropping a zero
Where it slips in: Because $\sqrt{49} = 7$, students tack on one zero and answer 7 or 700 instead of 70.
Don't do this: Guess the zero count. $4900$ has two trailing zeros, but the root gains only one.
The correct way: Use $4900 = 49 \times 100$, so $\sqrt{4900} = 7 \times 10 = 70$. Every pair of zeros in a perfect square becomes one zero in the root.
Mistake 3: Calling it irrational
Where it slips in: After a run of irrational roots like $\sqrt{47}$, the "it never ends" habit carries over to 4900.
Don't do this: Assume every square root is a messy decimal.
The correct way: Check whether the number is a perfect square first. When it factors into paired primes, as $2^2 \times 5^2 \times 7^2$ does, the root is a whole, rational number.
Conclusion
The square root of 4900 is exactly 70, since $70 \times 70 = 4900$.
4900 is a perfect square, so its root is a rational whole number, not a decimal.
Prime factorization gives $2^2 \times 5^2 \times 7^2$, and one factor from each pair multiplies to 70.
Grouping as $49 \times 100$ is the fastest route: $7 \times 10 = 70$.
The frequent errors are halving, mishandling the trailing zeros, and assuming the root is irrational.
For a teacher to walk through perfect squares and roots step by step, explore Bhanzu's algebra tutor or browse math classes online.
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