The square root of 48 is (4\sqrt{3}) in simplified radical form and approximately 6.9282 as a decimal. This guide explains what that answer means, how to simplify it step by step, why it is an irrational number, and how to use it in algebra, geometry, and everyday calculations.
Introduction
A square root is a number that produces a given value when multiplied by itself. In real terms, for example, (7 \times 7 = 49), so 7 is the square root of 49. Since 48 is close to 49 but is not a perfect square, its square root is slightly less than 7 and cannot be written as an exact terminating decimal Surprisingly effective..
The most useful exact answer is:
[ \sqrt{48}=4\sqrt{3} ]
Its decimal approximation is:
[ \sqrt{48}\approx 6.92820323 ]
Both forms are correct, but they serve different purposes. The radical form is exact, while the decimal form is convenient for measurement and estimation.
What Does the Square Root of 48 Mean?
The expression (\sqrt{48}) asks: “Which nonnegative number, when squared, equals 48?”
If we square the decimal approximation, we can check the result:
[ 6.92820323^2 \approx 48 ]
The result is approximately 48 because the displayed decimal has been rounded. The exact value continues without ending or repeating It's one of those things that adds up. That's the whole idea..
There is also a useful geometric interpretation. If a square has an area of 48 square units, its side length is:
[ \sqrt{48}=4\sqrt{3}\text{ units} ]
Because of this, each side measures approximately 6.93 units.
Step-by-Step Simplification
To simplify (\sqrt{48}), look for the largest perfect-square factor of 48.
Step 1: Find factors of 48
Some factors of 48 are:
- 1
- 2
- 3
- 4
- 6
- 8
- 12
- 16
- 24
- 48
The perfect-square factors are 1, 4, and 16. The largest is 16, making it the most efficient choice And that's really what it comes down to. That alone is useful..
Step 2: Rewrite 48 as a product
Separate 48 into 16 and another factor:
[ 48=16 \times 3 ]
This changes the radical to:
[ \sqrt{48}=\sqrt{16 \times 3} ]
Step 3: Separate the square root
Using the product rule (\sqrt{ab}=\sqrt{a}\sqrt{b}):
[ \sqrt{16 \times 3}=\sqrt{16}\times\sqrt{3} ]
Step 4: Simplify the perfect square
Because (\sqrt{16}=4):
[ \sqrt{16}\times\sqrt{3}=4\sqrt{3} ]
Thus:
[ \boxed{\sqrt{48}=4\sqrt{3}} ]
The number 3 has no perfect-square factor other than 1, so (4\sqrt{3}) is fully simplified.
Simplifying with Prime Factorization
Prime factorization provides another reliable method. First, break 48 into prime factors:
[ 48=2 \times 24 ]
[ 24=2 \times 12 ]
[ 12=2 \times 6 ]
[ 6=2 \times 3 ]
Therefore:
[ 48=2^4 \times 3 ]
A square root removes one factor from every pair of identical primes. The four factors of 2 form two pairs:
[ \sqrt{2^4 \times 3}=\sqrt{(2^2)^2 \times 3} ]
Taking the square root of ((2^2)^2), or 16, gives (2^2=4). The unpaired factor 3 remains inside the radical:
[ \sqrt{48}=4\sqrt{3} ]
This method is especially helpful when the largest perfect-square factor is not immediately obvious.
Decimal Approximation
Because 48 lies between the perfect squares 36 and 49:
[ 36<48<49 ]
its square root must lie between 6 and 7:
[ 6<\sqrt{48}<7 ]
Since 48 is much closer to 49 than to 36, (\sqrt{48}) is much closer to 7 than to 6.
A more accurate approximation is:
[ \sqrt{48}\approx
