Square Root Of 49: Value, Methods And Worked Examples

What Is A Square Root?

A square root of a number $n$ is a number $r$ with $r^2 = n$. For 49, both $7$ and $-7$ work, since $7^2 = 49$ and $(-7)^2 = 49$.

The radical sign $\sqrt{\phantom{x}}$ picks only the non-negative one, called the principal square root (the definition Wolfram MathWorld uses). So $\sqrt{49}$ means $7$, while "the square roots of 49" means $7$ and $-7$.

In exponent form, $\sqrt{49} = 49^{\frac{1}{2}} = 7$. The half power undoes squaring.

Where Does $\sqrt{49}$ Show Up, And Why Does It Matter?

The square root of 49 turns an area back into a length. A square garden, tile or screen with an area of 49 square units has sides of exactly 7 units.

It also hides inside one of the smaller Pythagorean triples, the 7, 24, 25 right triangle. If the hypotenuse is 25 and one leg is 24, the missing leg is $\sqrt{625 - 576} = \sqrt{49} = 7$.

Circles use it too. A circle whose area is $49\pi$ has radius $\sqrt{49} = 7$, which you get by undoing the $r^2$ in the area of a circle formula.

Quick Reference Table

Forty-nine sits between two other perfect squares, 36 and 64. The rows in between show how quickly the square roots of non-perfect squares turn into endless decimals.

For every value from 1 upwards, the square root 1 to 30 table lists the whole run in one place.

Is The Square Root Of 49 Rational Or Irrational?

The square root of 49 is rational. It equals $7$, and $7$ can be written as the fraction $\frac{7}{1}$, which is the test for a rational number.

Compare it with $\sqrt{48} \approx 6.9282$ or $\sqrt{50} \approx 7.0711$. Their decimals never end and never repeat, so each one is an irrational number.

Is 49 A Perfect Square?

Yes. A whole number is a perfect square when it equals some integer times itself, and $49 = 7 \times 7$.

That one fact settles everything else. The square root of a whole number is rational exactly when the number is a perfect square, so 49 lands among the perfect squares like 36, 64 and 81.

Why Is The Square Root Of 49 Rational When $\sqrt{50}$ Is Not?

The prime factors decide it. In $49 = 7^2$ every prime appears an even number of times, so the factors split into two equal halves.

In $50 = 2 \times 5^2$, the prime 2 appears once. No integer can supply half of a lone 2, which is why $\sqrt{50} = 5\sqrt{2}$ keeps a radical.

How Do You Find The Square Root Of 49?

Four hand methods give the square root of 49, and a calculator confirms it. Each one reaches $7$ by a different road.

Method 1: Prime Factorization

Divide 49 by the smallest prime that fits.

1. $49 \div 7 = 7$

2. $7 \div 7 = 1$

So the prime factorization is:

$$49 = 7 \times 7 = 7^2$$

Take one factor from each matching pair:

$$\sqrt{49} = \sqrt{7 \times 7} = 7$$

Method 2: Long Division

The long division method for square roots works digit pair by digit pair. For 49 there is only one pair, so the whole job takes one round.

1. Pair the digits from the right: $49$ is a single pair.

2. Find the largest digit $d$ with $d \times d \leq 49$. Since $6 \times 6 = 36$ and $8 \times 8 = 64$, the digit is $7$.

3. Write $7$ in the quotient and $7$ as the divisor, then subtract $7 \times 7 = 49$.

4. The remainder is $49 - 49 = 0$, so the division stops.

$$\sqrt{49} = 7 \text{ (remainder } 0)$$

A remainder of zero is the signal that 49 is a perfect square. For a number like 50 the remainder would be 1, and you would bring down pairs of zeros to keep going into decimals.

Method 3: Repeated Subtraction Of Odd Numbers

Subtract the odd numbers $1, 3, 5, 7, \ldots$ in order until you reach zero. The number of subtractions is the square root.

Seven subtractions bring 49 to zero, so $\sqrt{49} = 7$. This works because the sum of odd numbers from 1 up to the $n$th odd number is $n^2$, a pattern Wikipedia's square number article illustrates with growing dot squares.

Method 4: Units Digit And Estimation

This is the quickest mental check, and it scales to bigger perfect squares. Look at the last digit first, then bracket the number.

1. 49 ends in 9. Only numbers ending in 3 or 7 have squares ending in 9 ($3^2 = 9$, $7^2 = 49$).

2. Bracket it: $6^2 = 36 < 49 < 64 = 8^2$, so the root lies between 6 and 8.

3. The only candidate between 6 and 8 that ends in 3 or 7 is $7$, and $7^2 = 49$ confirms it.

Calculator Check

Any calculator gives $\sqrt{49} = 7$ with no decimal tail. If you want to see an approximation method settle on 7, start Newton's method at a guess of 8: $\frac{1}{2}\left(8 + \frac{49}{8}\right) = 7.0625$, and one more round gives about $7.0003$.

Why Is The Square Root Of 49 7 And Not ±7?

The symbol $\sqrt{49}$ names one number, and that number is $7$. The equation $x^2 = 49$ is a different question with two answers, $x = 7$ and $x = -7$.

Mixing the two is where many errors in quadratic equations begin. So write $\sqrt{49} = 7$, but write $x = \pm 7$ when you solve $x^2 = 49$.

What about $\sqrt{-49}$? No real number squares to a negative, so in the real numbers it has no value.

In complex numbers, $\sqrt{-49}$ equals $7i$, a topic that arrives later in algebra.

How Does $\sqrt{49}$ Help With $\sqrt{4900}$, $\sqrt{0.49}$ And $\sqrt{\frac{49}{9}}$?

Once you know $\sqrt{49} = 7$, a whole family of roots follows by splitting products and fractions. The rule is $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ for non-negative $a$ and $b$.

The full working for 4900 has its own page on the square root of 4900. The same shortcut is the reason $\sqrt{64} = 8$ makes $\sqrt{6400} = 80$, as the square root of 64 page shows.

Examples Of Square Root Of 49

Example 1

Evaluate $3\sqrt{49} - 5$.

$$3\sqrt{49} - 5 = 3 \times 7 - 5$$

$$= 21 - 5$$

$$= 16$$

Example 2

Find $\sqrt{49 + 576}$.

A tempting first move is to take each root separately: $\sqrt{49} + \sqrt{576} = 7 + 24 = 31$.

Check it by squaring: $31^2 = 961$, not $625$. The square root does not split across a plus sign.

Add first, then take the root:

$$\sqrt{49 + 576} = \sqrt{625}$$

$$= 25$$

This is the 7-24-25 triangle again, with 25 as the hypotenuse.

Example 3

Solve $x^2 - 49 = 0$.

$$x^2 = 49$$

$$x = \pm\sqrt{49}$$

$$x = 7 \text{ or } x = -7$$

Both values check: $7^2 - 49 = 0$ and $(-7)^2 - 49 = 0$.

Example 4

Find $\sqrt{0.49}$ without a calculator.

$$\sqrt{0.49} = \sqrt{\frac{49}{100}}$$

$$= \frac{7}{10}$$

$$= 0.7$$

The first instinct here is often $0.07$, because "the root of 49 is 7" gets attached to the last two decimal places. Writing the decimal as a fraction over 100 first removes the guesswork.

Example 5

Simplify $\sqrt{\frac{49}{9}}$ and give the decimal to 4 places.

$$\sqrt{\frac{49}{9}} = \frac{\sqrt{49}}{\sqrt{9}}$$

$$= \frac{7}{3}$$

$$\approx 2.3333$$

Example 6

A square garden has an area of 49 square metres. How much fencing goes around it?

$$\text{side} = \sqrt{49} = 7 \text{ m}$$

$$\text{perimeter} = 4 \times 7 = 28 \text{ m}$$

The garden needs 28 metres of fencing.

What Mistakes Do Students Make With The Square Root Of 49?

Mistake 1: Halving Instead Of Rooting

Where it slips in: Quick mental work, when "square root" gets read as "split in two".

Don't do this: Writing $\sqrt{49} = 24.5$, which is $49 \div 2$.

The correct way: Ask which number times itself gives 49. The rusher who halves never checks, yet $24.5 \times 24.5 = 600.25$, nowhere near 49.

Mistake 2: Writing $\sqrt{49} = \pm 7$

Where it slips in: Right after learning that 49 has two square roots.

Don't do this: Putting $\pm$ in front of the radical answer itself.

The correct way: The radical sign means the principal root, so $\sqrt{49} = 7$. Students who memorise "every positive number has two roots" tend to attach the $\pm$ everywhere; it belongs only to the solutions of $x^2 = 49$.

Mistake 3: Splitting The Root Across Addition

Where it slips in: Expressions such as $\sqrt{49 + 16}$ or $\sqrt{x^2 + 49}$.

Don't do this: Writing $\sqrt{49 + 16} = 7 + 4 = 11$.

The correct way: Add inside the radical first: $\sqrt{65} \approx 8.0623$. The product rule $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ is real, but there is no matching rule for sums, and that asymmetry is the habit worth building early.

Where Can You Get Help With Square Roots And Algebra?

Square roots of perfect squares are the first step toward radicals, irrational numbers and quadratic equations. If those later topics feel shaky, a teacher who works through the reasoning with you makes the difference.

You can work one-to-one with an algebra tutor, get targeted help with algebra on a single chapter, book a middle school math tutor for grades 6 to 8, or join math classes online for a steady weekly routine.

Conclusion

• The square root of 49 is exactly 7, because $7 \times 7 = 49$.

• 49 is a perfect square ($7^2$), so its root is rational and needs no simplifying.

• Prime factorization, long division, odd-number subtraction and the units-digit check all land on 7.

• $\sqrt{49}$ means only 7; the equation $x^2 = 49$ has the two solutions $\pm 7$.

• Knowing $\sqrt{49}$ unlocks $\sqrt{4900} = 70$, $\sqrt{0.49} = 0.7$ and $\sqrt{\frac{49}{9}} = \frac{7}{3}$.

Practice These Before Moving On

1. Evaluate $\sqrt{49} + \sqrt{81}$.

2. Use $\sqrt{49}$ and $\sqrt{4}$ to find $\sqrt{196}$.

3. Compute $\left(\sqrt{49}\right)^3$.

4. Solve $x^2 = \frac{49}{25}$.

5. A square tile has an area of $0.49$ square metres. How long is each side?

Answers:

• Answer to Question 1: $16$

• Answer to Question 2: $14$

• Answer to Question 3: $343$

• Answer to Question 4: $x = \frac{7}{5}$ or $x = -\frac{7}{5}$

• Answer to Question 5: $0.7$ m

If Question 4 gave you only one answer, go back to the section on why $\sqrt{49}$ is 7 and not $\pm 7$. Want a teacher to check your working? Book a free demo class.

Read More

• Squares and square roots explained from the start

• Square root tricks for quick mental estimates

• Square root of 50 in simplest radical form

• Square root of 47 and how to estimate it

• Simplifying radical expressions step by step