Your calculator shows 4.242640687, but the answer key gives . Did you make a mistake? Probably not. Both answers can describe the square root of 18, although the displayed decimal is rounded. Understanding exact form vs decimal form helps you recognize equivalent answers and decide what to write in your working.
What an exact answer tells you
An exact answer preserves the full value. For the square root of 18, factor 18 into 9 times 2. The square root of 9 is 3, so . No rounding has taken place.
The decimal expansion continues without terminating or repeating. Writing 4.24 gives an approximation to two decimal places; it does not capture the entire value. Use the approximation symbol when showing this relationship: .
Not every square root needs a radical in its final answer. The square root of 81 is exactly 9. Decimals can also be exact: the square root of 2.25 is exactly 1.5. The issue is whether any digits have been discarded, not whether a decimal point appears.
Remember that the square-root symbol means the principal, nonnegative root when working with real numbers. The square root of 81 is 9, not negative 9. Solving a separate equation such as x squared equals 81 has two solutions, 9 and negative 9. Reading the question carefully prevents that common mix-up.
See how early rounding changes a result
Suppose a textbook problem gives a square an exact area of 18 square centimetres. Its side length is centimetres. Multiplying by four gives an exact perimeter of
centimetres, approximately 16.97 centimetres.
If you round the side to 4.2 centimetres first, the calculated perimeter becomes 16.8 centimetres. That difference comes from rounding an intermediate value. Keeping the radical until the final step avoids this loss of precision.
To compare the two forms, enter 18 with root index 2 in Radical Calculator. Its exact and decimal outputs let you check the relationship after working through the factorization yourself.
Recognize the same value in exponent notation
A fractional exponent can express a root without the radical symbol. For example, means the square root of 18. It has exactly the same value as
.
For positive bases, the denominator of a reduced fractional exponent specifies the root, while the numerator specifies the power. Therefore, can be written as
. Taking the cube root of 64 and then squaring gives 16.
If the notation is unfamiliar, use the radical form conversion tool to check how the exponent becomes a radical. This checks the rewrite; finding the numerical value is a separate step. Keep any original parentheses around a grouped base.
Choose the form the question asks for
When a question asks for an exact answer, retain the simplified radical unless the root evaluates exactly to a rational number. When it specifies decimal places or significant figures, carry the exact value through your working and round the final answer accordingly.
Try this check: a square has an exact area of 28 square metres. Its side is metres, approximately 5.29 metres. Squaring the exact expression returns 28, while squaring the rounded decimal gives 27.9841. That small mismatch is a reminder that an approximation represents a nearby value.
