What is the square root of 27?

The number 27 is not a perfect square. So the answer is not a simple whole number like 4 or 5. To find the exact answer, you must simplify it.

You can think of what numbers make 27. It is 9 times 3. We know the square root of 9 is 3. This is easy. So, you can take the 3 outside of the square root symbol. The other 3 must stay inside.

So the most exact answer is 3√3.

If you need this as a decimal for your homework, it is an irrational number. The numbers go on forever. On a calculator, the answer is approximately 5.196. I hope this is a help for you.

The square root of 27 is an irrational number, which means it cannot be expressed as a simple fraction and its decimal representation goes on forever without repeating. The simplified radical form is 3√3. Here's how we get there: First, we find the prime factorization of 27, which is 3 × 3 × 3. We can pair the prime factors into groups of two; one pair of 3s comes out of the radical as a single 3, and the remaining unpaired 3 stays under the radical. Therefore, √27 = √(9 × 3) = √9 × √3 = 3√3. This is the exact value. If you need a decimal approximation, 3√3 is approximately 5.196. It's important to understand the difference between the exact form (3√3) and the decimal approximation, as the exact form is often required in mathematical solutions to maintain precision.

From a more conceptual perspective, the square root of a number asks: "What number, when multiplied by itself, gives me this number?" So for 27, we're looking for a value x such that x × x = 27. We know that 5 × 5 = 25 and 6 × 6 = 36, so the answer must be between 5 and 6. The fact that 27 is not a perfect square (like 25 or 36) tells us immediately that its square root will be irrational. This irrationality is why we often represent it exactly using the radical symbol. The simplified form, 3√3, is not just a random arrangement; it's a precise mathematical expression. It's also worth noting that in geometry, if you had a square with an area of 27 square units, the length of each side would be √27, or 3√3, units long.

Let's get practical. Why would you even need to know this? Well, 3√3 appears all the time in trigonometry and geometry. For example, it's the exact length of the long side (the hypotenuse) of a 30-60-90 right triangle where the short side is 3. It's also found in calculations involving the volume of cubes or in the Pythagorean theorem. While the decimal approximation (5.196) is useful for real-world measurement (like cutting a piece of wood), the exact form (3√3) is crucial for deriving formulas and ensuring that subsequent calculations don't accumulate rounding errors. So, knowing both the simplified radical and its decimal value is important, depending on whether you're in a math class (use the radical) or on a construction site (use the decimal).

It's also important to note that every positive number has two square roots: one positive and one negative. So, while we primarily focus on the principal (positive) square root, √27 = 3√3, it is equally true that -3√3 is also a square root of 27, because (-3√3) × (-3√3) = 9 × 3 = 27. This concept is fundamental in algebra, especially when solving quadratic equations like x² = 27. The solutions would be x = √27 and x = -√27, or x = 3√3 and x = -3√3. Forgetting the negative root is a common mistake, so always remember that the radical symbol (√) itself denotes only the principal (non-negative) square root, but the equation x² = a has two solutions.

Here's a fun historical and computational angle. Before calculators, finding square roots was a tedious process involving algorithms like the "Babylonian method" or using lookup tables. Let's try the Babylonian method for √27. You make a guess. Since 5²=25 is close, let's guess 5.2. Then, you divide 27 by your guess: 27 / 5.2 ≈ 5.192. Now, average this result with your guess: (5.2 + 5.192)/2 = 5.196. This is your new, better guess. If you repeat the process: 27 / 5.196 ≈ 5.19615. The average is still about 5.196. You can see how quickly it converges to the value we know! This ancient algorithm, thousands of years old, is effectively what your calculator does at lightning speed to give you that decimal approximation.

As a math tutor, I always tell my students that understanding how to simplify a square root is more important than just the answer. Let's break down √27. Look for the largest perfect square that is a factor of 27. The perfect squares are 1, 4, 9, 16, 25... 9 is a factor of 27 (9 × 3 = 27). So, we can rewrite the problem: √27 = √(9 × 3). The square root of 9 is 3, so we can take that out of the radical, leaving us with 3√3. This is the simplest radical form. Remember, the goal of simplification is to get the number under the radical as small as possible. If you're using a calculator, you'll just get the decimal: about 5.1961524227. But for most algebra work, leaving it as 3√3 is the correct and preferred answer.