What Is A Square Root?
A square root of a number $n$ is a value $r$ with $r^2 = n$. The square root of 676 is the number that, multiplied by itself, gives 676.
Here the search ends on a whole number: $26 \times 26 = 676$. That clean landing is what makes 676 special.
Where Does √676 Appear?
$\sqrt{676} = 26$ is the hypotenuse of the right triangle with legs 10 and 24, one of the classic Pythagorean triples, since $10^2 + 24^2 = 100 + 576 = 676$. It also appears as the side of a square whose area is 676 square units, which comes out to a clean 26 units.
Quick Reference Table
Number $n$ | $\sqrt{n}$ | Perfect Square? |
|---|---|---|
576 | 24 | Yes |
625 | 25 | Yes |
676 | 26 | Yes |
729 | 27 | Yes |
784 | 28 | Yes |
700 | 26.458 | No |
Is The Square Root Of 676 Rational Or Irrational?
$\sqrt{676}$ is rational - in fact it is the whole number 26, which can be written as the fraction $\frac{26}{1}$.
A whole number has a rational root exactly when it is a perfect square, and 676 qualifies. $$676 = 2^2 \times 13^2 = (2 \times 13)^2 = 26^2$$
Every prime appears an even number of times, so the root is a rational number. Compare this with the many non-perfect-squares in the square root 1 to 30 list, whose roots run on forever.
How Do You Find √676? (Prime Factorisation And Long Division)
Prime factorisation is the cleanest method for a perfect square. Break 676 into primes and pair them.
Step 1: Divide by 2 twice. $$676 = 2 \times 338 = 2 \times 2 \times 169$$
Step 2: Factor 169. $$169 = 13 \times 13$$
Step 3: Take one prime from each pair. $$\sqrt{676} = \sqrt{2^2 \times 13^2} = 2 \times 13 = 26$$
Long division confirms it and works even when you do not spot the factors, just like ordinary long division.
Step 1: Pair the digits. $$\overline{6}\ \overline{76}$$
Step 2: The largest square at most 6 is 4. $$2^2 = 4$$
Step 3: Subtract and bring down 76. $$6 - 4 = 2 \rightarrow 276$$
Step 4: Double the quotient 2 to 4, then find $d$ with $(40 + d)\times d \le 276$. $$46 \times 6 = 276$$
Step 5: The remainder is zero, so the root is exact. $$\sqrt{676} = 26$$
Examples Of √676
Example 1
Verify that $\sqrt{676} = 26$.
Square the proposed answer. $$26^2 = 26 \times 26 = 676$$
Since squaring 26 returns 676, the square root of 676 is 26.
Example 2
A student says $\sqrt{676} = 338$ because $676 \div 2 = 338$. Why is that wrong?
Look at the wrong path. The student halved 676 instead of taking its square root.
Test the claim. $$338^2 = 114244$$
That is far larger than 676, so 338 cannot be the root.
The correct idea: a square root asks "what times itself gives 676?", not "what is half of 676?". The answer is $26$, since $26 \times 26 = 676$.
Example 3
Find $\sqrt{676}$ using its prime factorisation.
Factor into primes. $$676 = 2^2 \times 13^2$$
Take one factor from each pair. $$\sqrt{676} = 2 \times 13 = 26$$
Example 4
A square field has an area of 676 square metres. Find the length of one side.
The side is the square root of the area. $$s = \sqrt{676} = 26 \text{ m}$$
Each side is exactly 26 metres, with no decimals.
Example 5
Evaluate $\sqrt{676} - \sqrt{625}$.
Both numbers are perfect squares. $$\sqrt{676} = 26$$ $$\sqrt{625} = 25$$
Subtract. $$26 - 25 = 1$$
Common Mistakes
Mistake 1: Dividing instead of rooting
Where it slips in: Confusing "square root of 676" with "676 divided by 2".
Don't do this: Writing $\sqrt{676} = 338$.
The correct way: A root asks what number squared gives 676, which is 26, since $26 \times 26 = 676$. The learners who reach for division here never test their answer by squaring it.
Mistake 2: Forgetting the negative root in equations
Where it slips in: Solving $x^2 = 676$ rather than evaluating $\sqrt{676}$.
Don't do this: Writing $\sqrt{676} = \pm 26$ for the principal root.
The correct way: The symbol $\sqrt{676}$ means the positive root, 26. The equation $x^2 = 676$ has both solutions, $x = 26$ and $x = -26$.
Mistake 3: Misreading the prime factors
Where it slips in: Pairing the primes of 676 too quickly.
Don't do this: Taking $\sqrt{676} = 2 \times 2 \times 13 = 52$ by keeping both 2s.
The correct way: Take one prime from each pair, so $\sqrt{2^2 \times 13^2} = 2 \times 13 = 26$.
Conclusion
The square root of 676 is exactly 26, a rational whole number.
676 is a perfect square: $676 = 2^2 \times 13^2 = 26^2$.
Prime factorisation and long division both land on 26 with no remainder.
A square root is not the same as dividing by 2, the trap in Example 2.
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