# 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

| Number | Square root (approx.) | Exact / simplified | Rational? |
| --- | --- | --- | --- |
| 36 | 6.000 | 6 | Rational |
| 40 | 6.325 | \( 2\sqrt{10} \) | Irrational |
| 43 | 6.557 | \( \sqrt{43} \) | Irrational |
| 45 | 6.708 | \( 3\sqrt{5} \) | Irrational |
| 49 | 7.000 | 7 | Rational |
| 153 | 12.369 | \( 3\sqrt{17} \) | Irrational |
| 255 | 15.969 | \( \sqrt{255} \) | Irrational |

## 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.