Square Root of 43 — Value, Simplification, and Steps

The square root of 43 is approximately 6.557, and it stays as ( \sqrt{43} ) because 43 is prime and has no perfect-square factor.

Quick Answer:
Result: ( \sqrt{43} \approx 6.557 )
Notation: ( \sqrt{43} ) (already in simplest radical form)
Method shown: Long division and approximation
Approximate value (irrational): 6.5574385246
Exact form: ( \sqrt{43} ) (cannot be simplified further)

Quick Reference Table

Where The Square Root of 43 Appears

( \sqrt{43} ) is the length of the diagonal drawn across a box whose side-squares sum to 43, and it appears in distance calculations on a coordinate grid where the horizontal and vertical gaps square to 43. Because 43 is prime, the root refuses to simplify, which makes it a clean textbook example of an irrational that has to stay under the radical.

What The Square Root of 43 Means

The square root of 43 is the positive number whose square is 43. In symbols, ( \sqrt{43} \times \sqrt{43} = 43 ).

Because ( 6^2 = 36 ) and ( 7^2 = 49 ), the answer sits between 6 and 7, roughly halfway.

Is The Square Root of 43 Rational or Irrational?

The square root of 43 is irrational. A rational number can be written as a fraction ( \frac{p}{q} ) of two integers; ( \sqrt{43} ) cannot.

43 is a prime number, with only the factors 1 and 43, so it has no repeated prime factor and is not a perfect square. Its decimal expansion runs on forever without repeating.

How To Compute The Square Root of 43

Method 1: Long division

Since 43 is a two-digit number, treat it as one pair: 43.

Find the largest number whose square is at most 43: that is 6, since ( 6^2 = 36 ) and ( 7^2 = 49 ).

Subtract to get remainder 43 − 36 = 7; place a decimal point and bring down 000000 to make 700.
Double the quotient so far (6) to get 12; find a digit ( x ) so that ( 12x\times x \le 700 ).

Test ( x=5 ): ( 125\times5=625 \le 700 ); test ( x=6 ): ( 126\times6=756 > 700 ). So ( x=5 ).

The quotient is now 6.5, remainder 700 − 625 = 75; bring down 000000 to make 7500.

Continue the process to reach 6.55…

Final answer: ( \sqrt{43} \approx 6.557 )

Method 2: Estimation between perfect squares

43 lies between ( 6^2 ) and ( 7^2 ), so the root is between 6 and 7.

The gap from 36 to 43 is 7; the gap from 36 to 49 is 13.

Estimate the fraction: 6 + ( \frac{7}{13} \approx 6.54 ).

Refining with long division sharpens this to 6.557.

Final answer: ( \sqrt{43} \approx 6.557 )

Common Mistakes With The Square Root of 43

Mistake 1: Trying to simplify the root of a prime

Where it slips in: Assuming any number under a radical should reduce to ( a\sqrt{b} ).

Don't do this: Write ( \sqrt{43} ) as some product of a whole number and a smaller root.

The correct way: ( \sqrt{43} ) is already simplest.

Mistake 2: Rounding 43 up to 49 and reading 7

Where it slips in: Noticing 43 is fairly close to 49.

Don't do this: Write ( \sqrt{43} = 7 ).

The correct way: 49 ≠ 43, so 43 < 7; the value is 6.557.

Mistake 3: Placing the decimal point too early in long division

Where it slips in: After the first digit 6, before the remainder is fully handled.

Don't do this: Insert the decimal before bringing down the first pair of zeros.

The correct way: The decimal point in the quotient goes in only when you cross from the integer part into the fractional part.

A Quick Way To Check Yourself

Estimate first: 43 sits between 36 and 49; any answer outside that band is wrong. Then square your decimal back: ( 6.557^2 \approx 42.99 ), which confirms it.