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 $(\sqrt{448})^2 = 448$.
To write a root in simplest radical form, you pull out the largest perfect-square factor. The full method sits in the guide to simplifying radical expressions, and the broader pattern of exact and inexact roots is set out in squares and square roots.
Where Does the Square Root of 448 Appear?
$\sqrt{448}$ appears as a length whenever a squared distance equals 448, for example the diagonal of a rectangle whose side-squares total 448, or a resultant in a Pythagorean setup that lands on 448. It also turns up when simplifying radicals in algebra, where keeping the exact form $8\sqrt{7}$ rather than the decimal $\approx 21.166$ preserves accuracy through the rest of a calculation.
Quick Reference Table
The table shows $\sqrt{448}$ inside the $\sqrt{7}$ family, where each radicand is a perfect square times 7. Only 441 and 484 give whole-number roots.
Number | Square Root | Type |
|---|---|---|
$\sqrt{441}$ | $21$ | Rational (perfect square) |
$\sqrt{448}$ | $8\sqrt{7} \approx 21.166$ | Irrational |
$\sqrt{484}$ | $22$ | Rational (perfect square) |
$\sqrt{112}$ | $4\sqrt{7} \approx 10.583$ | Irrational ($\sqrt{7}$ family) |
$\sqrt{252}$ | $6\sqrt{7} \approx 15.875$ | Irrational ($\sqrt{7}$ family) |
$\sqrt{567}$ | $9\sqrt{7} \approx 23.812$ | Irrational ($\sqrt{7}$ family) |
$\sqrt{7}$ | $\approx 2.646$ | Irrational |
Is the Square Root of 448 Rational or Irrational?
$\sqrt{448}$ is irrational. It simplifies to $8\sqrt{7}$, and since $\sqrt{7}$ is irrational, any whole-number multiple of it is irrational too.
The reason is the leftover factor. $448 = 2^6 \times 7$, so the perfect-square part ($2^6 = 64$) comes out cleanly, but the single 7 stays under the radical with no pair. Because $\sqrt{7}$ has a non-terminating, non-repeating decimal, so does $8\sqrt{7}$.
How Do You Find the Square Root of 448?
Method 1: Prime factorization and simplification
Break 448 into prime factors.
$448 = 2^6 \times 7$
Separate the perfect-square part from the leftover.
$\sqrt{448} = \sqrt{2^6} \times \sqrt{7}$
$\sqrt{2^6} = 2^3 = 8$.
$\sqrt{448} = 8\sqrt{7}$.
Final answer: $\sqrt{448} = 8\sqrt{7}$
Method 2: Largest perfect-square factor
Find the largest perfect square that divides 448.
$448 = 64 \times 7$, and 64 is a perfect square.
$\sqrt{448} = \sqrt{64} \times \sqrt{7}$
$\sqrt{64} = 8$, so $\sqrt{448} = 8\sqrt{7}$.
Final answer: $\sqrt{448} = 8\sqrt{7}$
Method 3: Decimal estimation
Estimate $\sqrt{7}$ and scale it.
$\sqrt{7} \approx 2.6458$, and $21^2 = 441$ confirms $\sqrt{448}$ is just above 21.
$8 \times 2.6458 = 21.166$.
So $\sqrt{448} \approx 21.166$.
Final answer: $\sqrt{448} \approx 21.166$
Examples of the Square Root of 448
Example 1: Watch How This Goes Wrong
A student factors $448 = 4 \times 112$, writes $\sqrt{448} = 2\sqrt{112}$, and stops.
The radical is not fully simplified, because 112 still holds a perfect-square factor.
$\sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7}$, so $2\sqrt{112} = 2 \times 4\sqrt{7} = 8\sqrt{7}$.
Always pull out the largest perfect square, or simplify until none remains.
Example 2: Simplifying by Prime Factorization
Simplify $\sqrt{448}$ from scratch.
$448 = 2^6 \times 7$.
$\sqrt{448} = \sqrt{2^6} \times \sqrt{7} = 8\sqrt{7}$.
The simplest radical form is $8\sqrt{7}$.
Example 3: Checking the Decimal Value
Find $\sqrt{448}$ as a decimal.
$\sqrt{448} = 8\sqrt{7}$ and $\sqrt{7} \approx 2.6458$.
$8 \times 2.6458 = 21.166$.
So $\sqrt{448} \approx 21.166$.
Example 4: Verifying the Simplified Form
Confirm that $8\sqrt{7}$ squares back to 448.
$(8\sqrt{7})^2 = 8^2 \times (\sqrt{7})^2$.
$= 64 \times 7$.
$= 448$, which checks out.
Example 5: Multiplying √448 by Itself
Evaluate $\sqrt{448} \times \sqrt{448}$.
A square root times itself returns the radicand.
$\sqrt{448} \times \sqrt{448} = 448$.
Common Mistakes
Mistake 1: Stopping Before the Radical Is Fully Simplified
Where it slips in: A student pulls out a small square factor and leaves another one inside.
Don't do this: Write $\sqrt{448} = 2\sqrt{112}$ and treat it as the final answer.
The correct way: Keep factoring until no perfect square remains inside: $\sqrt{448} = 8\sqrt{7}$.
Mistake 2: Multiplying the Outside and Inside Numbers
Where it slips in: After reaching $8\sqrt{7}$, a student combines the 8 and the 7.
Don't do this: Write $8\sqrt{7} = \sqrt{56}$ by folding 8 back under the radical incorrectly.
The correct way: The 8 is a coefficient outside the radical; $8\sqrt{7}$ is fully simplified and equals about 21.166, while $\sqrt{56} \approx 7.48$.
Mistake 3: Confusing √448 With 448²
Where it slips in: Reading quickly, a student squares 448 instead of rooting it.
Don't do this: Answer 200704 for $\sqrt{448}$; that is $448^2$, the opposite operation.
The correct way: The square root asks what number times itself gives 448, which is $8\sqrt{7} \approx 21.166$.
Conclusion
The square root of 448 is $8\sqrt{7}$, approximately 21.166, and is irrational.
$448 = 2^6 \times 7$, so the perfect-square factor 64 pulls out as 8 and leaves $\sqrt{7}$.
It sits between 21 and 22, close to 21, because 448 is just above the perfect square 441.
Prime factorization and largest-square-factor methods both give $8\sqrt{7}$.
Keep the exact form $8\sqrt{7}$ through a calculation and convert to $\approx 21.166$ only at the end.
To master radical simplification with a teacher, explore Bhanzu's algebra tutor sessions or browse math classes online. Want a live Bhanzu trainer to walk through more radical problems? Book a free demo class.
For a plain-language walkthrough of pulling out square factors, see Math is Fun on simplifying square roots.
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