Question

Simplify the expression. $$\sqrt{216}$$

Short answer

The simplified form of \(\sqrt{216}\) is \(6\sqrt{6}\).

Step-by-step solution

Find prime factors of 216

Start by finding the prime factors of the given number (216). It's found that 216 = 2*2*2*3*3*3. So, it can be expressed as \(2^{3} \cdot 3^{3}\).

Simplify the square root

Next, the square root of 216 can be expressed as \(\sqrt{216} = \sqrt{2^{3} \cdot 3^{3}}\). We can break this up further into \(\sqrt{2^{3}} \cdot \sqrt{3^{3}}\) which then simplifies to \(2\sqrt{2} \cdot 3\sqrt{3}\).

Simplify the expression

Lastly, we just multiply like terms together to result in \(6\sqrt{6}\).

Key concepts

Prime Factorization

Prime factorization is a crucial process in arithmetic that involves breaking down a composite number into a product of its prime factors. Primes are the building blocks of all natural numbers and are defined as numbers greater than 1 that have no divisors other than 1 and themselves. To perform prime factorization, one typically starts with the smallest prime number, 2, and continues dividing the original number by prime numbers until the result is a prime number itself.

For example, let's take the number 216 as in the exercise. We start by dividing 216 by 2, the smallest prime number, and keep dividing by 2 until we cannot do so evenly. Then we proceed with the next smallest prime number, which is 3. So, 216 can be broken down into 2 x 108, then 2 x 2 x 54, and further until we end up with 2 x 2 x 2 x 3 x 3 x 3, which is the prime factorization of 216, often written as \(2^{3} \cdot 3^{3}\). Knowing the prime factors is essential in simplifying radicals as it reveals pairs of prime numbers, crucial for the simplification process.

Radicals Simplification

Radicals simplification involves reducing expressions containing roots to their simplest form. When simplifying square roots, the objective is to look for prime factors that appear in pairs, as a pair of identical factors can be taken out of the radical sign as a single number. This is because the square root of any number is a value that, when multiplied by itself, gives the original number, and a pair of prime factors does just that.

So, the square root of \(2^{3} \cdot 3^{3}\) from our example is split into \(\sqrt{2^{3}} \cdot \sqrt{3^{3}}\). Since there are pairs for both 2 and 3 within the radicals, we take out a single 2 and a single 3 and leave the remainder inside the roots. This results in \(2\sqrt{2} \cdot 3\sqrt{3}\). Simplifying further, we look for any potential simplifications inside the roots or by multiplying any coefficients outside, leading us to our final simplified form.

Square Root Operations

Square root operations entail performing arithmetic operations, such as multiplication, division, addition, or subtraction, on expressions containing square roots. The fundamental rule to remember is that the square root of a product of two numbers is the product of the square roots of these numbers, and similarly, the square root of a quotient is the quotient of the square roots. However, this does not apply to addition and subtraction, where simplification rules depend on like terms.

When we multiply the simplified forms from our previous example, we are engaging in square root multiplication: \(2\sqrt{2} \cdot 3\sqrt{3}\). Multiplying the coefficients (2 and 3) and multiplying the radicals \(\sqrt{2} \cdot \sqrt{3}\), we arrive at the solution, \(6\sqrt{6}\). It's essential to handle square root expressions with care, especially when dealing with more complicated operations involving addition or subtraction of radicals, where like terms can only be combined.