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): $6.557438524$

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 $00$ to make $700$.

Double the quotient so far ($6$) to get $12$; find a digit $x$ so that $12x \times x \le 700$.

Test $x = 5$: $125 \times 5 = 625 \le 700$; test $x = 6$: $126 \times 6 = 756 > 700$. So $x = 5$.

The quotient is now $6.5$, remainder $700 - 625 = 75$; bring down $00$ to make $7500$.

Continue the process to reach $6.55\ldots$

Final answer: $\sqrt{43} \approx 6.557$

Method 2: Estimation between perfect squares

$43$ lies between $36 = 6^2$ and $49 = 7^2$, 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: $6 + \dfrac{7}{13} \approx 6.54$.

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: $43$ is prime, so it has no perfect-square factor; $\sqrt{43}$ is already simplest. The first instinct is to look for a factor to extract, but the root of a prime never simplifies.

Mistake 2: Rounding 43 up to 49 and reading 7

Where it slips in: Noticing $43$ is fairly close to $49 = 7^2$.

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

The correct way: $7^2 = 49 \ne 43$, so $\sqrt{43} < 7$; 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 — after subtracting $36$ and bringing down $00$.

A Quick Way To Check Yourself

Estimate first: $43$ sits between $36$ and $49$, so the root is between $6$ and $7$, and any answer outside that band is wrong. Then square your decimal back: $6.557^2 \approx 42.99$, which confirms it. To build square-root fluency with a teacher, explore Bhanzu's algebra tutor or structured help with algebra.

Read More

• Square root — what a square root is and how the symbol works.

• Square root 1 to 25 — the reference chart for the small roots.

• Square root of 153 — simplifies to $3\sqrt{17}$.

• Square root of 567 — simplifies to $9\sqrt{7}$.

• Squares and square roots — the wider topic.