Square Root of 79 - Value, Simplest Form & Steps

What Is a Square Root?

A square root of a number is a value that multiplies by itself to give that number. Squaring and taking the square root are inverse operations, so $(\sqrt{79})^2 = 79$.

Because 79 is not a perfect square, no whole number squares to it. Since $8^2 = 64$ and $9^2 = 81$, the value $\sqrt{79}$ sits between 8 and 9, closer to 9. If you want the full pattern of exact and inexact roots, the guide to squares and square roots lays them out together.

Where Does the Square Root of 79 Appear?

$\sqrt{79}$ shows up as a length whenever a squared distance equals 79, for example the hypotenuse of a right triangle whose leg-squares sum to 79, or the radius of a circle whose area is $79\pi$. It also appears in algebra as the exact solution of $x^2 = 79$, where the answer must stay as $\pm\sqrt{79}$ and only convert to $\approx 8.888$ at the final step to avoid rounding error.

Quick Reference Table

The table places $\sqrt{79}$ among its neighbours. Only the perfect square 81 gives a whole-number root, so every other row is irrational.

Is the Square Root of 79 Rational or Irrational?

$\sqrt{79}$ is irrational. A rational number can be written as a ratio of two integers, but $\sqrt{79}$ cannot, and its decimal is non-terminating and non-repeating.

The reason is the factorisation. 79 is a prime number, so it factors only as $1 \times 79$. With no repeated prime factor, there is no pair to pull out from under the radical, and no whole-number root exists. The same contradiction argument used to prove that root 7 is irrational applies to every non-perfect-square whole number, including 79.

How Do You Find the Square Root of 79?

Method 1: Prime factorization (why it will not simplify)

Factor 79 into primes.

$79 = 79$ (79 is prime)

There is no repeated factor, so no perfect square can be separated.

$\sqrt{79}$ stays as $\sqrt{79}$.

Final answer: $\sqrt{79}$ is already in simplest radical form.

Method 2: Estimation between perfect squares

Find the two nearest perfect squares.

$8^2 = 64$ and $9^2 = 81$, so $8 < \sqrt{79} < 9$.

Since 79 is very close to 81, the root is close to 9.

Test 8.88: $8.88^2 = 78.8544$, slightly low.

Test 8.89: $8.89^2 = 79.0321$, slightly high.

So $\sqrt{79} \approx 8.888$.

Method 3: Long division

Write 79 as $\overline{79}.\overline{00}\ \overline{00}$ and pair the digits around the decimal point.

The largest square under 79 is 64, and $\sqrt{64} = 8$, so the first digit is 8; remainder $79 - 64 = 15$.

Bring down $00$ to make 1500, then double 8 to get 16, and find $d$ with $(160 + d) \times d \le 1500$: $d = 8$ gives $168 \times 8 = 1344$.

The quotient is 8.8; remainder $1500 - 1344 = 156$, so bring down $00$ to make 15600.

Double 88 to get 176, and find $d$ with $(1760 + d) \times d \le 15600$: $d = 8$ gives $1768 \times 8 = 14144$.

The quotient reads $8.88\ldots$, refining to $\approx 8.888$.

Final answer: $\sqrt{79} \approx 8.888$

Examples of the Square Root of 79

Example 1: The Tempting Shortcut That Doesn't Work

A student assumes every root simplifies and writes $\sqrt{79} = \sqrt{4} \times \sqrt{something}$.

$79$ has no perfect-square factor, since it is prime.

So nothing can be pulled out, and $\sqrt{79}$ is already simplest.

The only "simpler" form is the decimal approximation $\approx 8.888$, which is not exact.

Example 2: Placing √79 Between Two Integers

Find the nearest perfect squares below and above 79.

$8^2 = 64$ and $9^2 = 81$.

Since $64 < 79 < 81$, it follows that $8 < \sqrt{79} < 9$.

$\sqrt{79} \approx 8.888$, closer to 9.

Example 3: Multiplying √79 by Itself

Evaluate $\sqrt{79} \times \sqrt{79}$.

A square root times itself returns the radicand.

$\sqrt{79} \times \sqrt{79} = 79$.

Example 4: Solving x² = 79

Solve $x^2 = 79$ for $x$.

Take the square root of both sides.

$x = \pm\sqrt{79}$.

$x \approx 8.888$ or $x \approx -8.888$.

Example 5: Comparing √79 and √80

Decide which is larger, $\sqrt{79}$ or $\sqrt{80}$.

The square-root function increases, so a larger radicand gives a larger root.

$79 < 80$, so $\sqrt{79} < \sqrt{80}$.

Numerically, $8.888 < 8.944$.

Common Mistakes

Mistake 1: Trying to Simplify the Radical

Where it slips in: A student expects every root to reduce, so they hunt for a factor to pull out of $\sqrt{79}$.

Don't do this: Write $\sqrt{79}$ as a product of two smaller radicals hoping one is a perfect square.

The correct way: Check for square factors first. 79 is prime, so $\sqrt{79}$ is already the simplest exact form; only its decimal, $\approx 8.888$, is an approximation.

Mistake 2: Rounding Too Early

Where it slips in: In a longer calculation, a student replaces $\sqrt{79}$ with 8.9 in step one.

Don't do this: Carry the rounded 8.9 through every later step and treat the result as exact.

The correct way: Keep $\sqrt{79}$ in radical form through the algebra and convert to $\approx 8.888$ only at the end.

Mistake 3: Confusing √79 With 79²

Where it slips in: Reading quickly, a student squares 79 instead of rooting it.

Don't do this: Answer 6241 for $\sqrt{79}$; that is $79^2$, the opposite operation.

The correct way: The square root asks what number times itself gives 79, which is about 8.888, not 6241.

Conclusion

• The square root of 79 is approximately 8.888 and is irrational.

• 79 is prime, so $\sqrt{79}$ has no square factor and is already in simplest radical form.

• It sits between 8 and 9, closer to 9, because 79 is near the perfect square 81.

• Estimation and long division both reach $\approx 8.888$.

• Keep $\sqrt{79}$ exact through a calculation and round only at the end.

To take irrational roots further with a teacher, explore Bhanzu's algebra tutor sessions or browse math classes online. Want a live Bhanzu trainer to walk through more square-root problems? Book a free demo class.

For a formal treatment of the square-root operation, the Wolfram MathWorld entry on square roots is a reliable reference.

Read More

• Square roots from 1 to 30, a full reference table

• Square root of 78, the neighbour just below

• Square root of 68, another irrational root

• Square root tricks for fast estimation

• Simplifying radical expressions step by step

• Perfect squares and how to recognise them