Question

Simplify: \(\sqrt{540}\)

Short answer

The simplified form of \(\sqrt{540}\) is \(6 \sqrt{15}\).

Step-by-step solution

- Prime Factorization

Break down the number 540 into its prime factors. Start by dividing 540 by the smallest prime number, which is 2: \[ 540 \div 2 = 270 \] Continue dividing by 2: \[ 270 \div 2 = 135 \] Since 135 is not divisible by 2, move to the next prime number, which is 3:\[ 135 \div 3 = 45 \] Continue dividing by 3: \[ 45 \div 3 = 15 \] And again by 3:\[ 15 \div 3 = 5 \] Finally, 5 is a prime number. So the prime factorization of 540 is: \[ 540 = 2^2 \times 3^3 \times 5 \]

- Pair the Prime Factors

Group the prime factors in pairs to simplify under the square root: \[ \sqrt{540} = \sqrt{2^2 \times 3^3 \times 5} \]

- Simplify the Square Root

Take the square root of the paired prime factors: \[ \sqrt{2^2 \times 3^3 \times 5} = \sqrt{(2^2) \times (3^2 \times 3) \times 5} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{3 \times 5} \]Simplify further: \[ 2 \times 3 \times \sqrt{15} \]So, \[ \sqrt{540} = 6 \sqrt{15} \]

Key concepts

Prime Factorization

Prime factorization is breaking down any composite number into its prime numbers. For example, to simplify the square root of 540, we first need to find its prime factors. Start with the smallest prime number. Here, it is 2.

• Divide 540 by 2: \( 540 \div 2 = 270 \)
• Continue this until it doesn't divide evenly: \[ 270 \div 2 = 135 \]

Since 135 is not divisible by 2, move to 3, the next smallest prime number.

• Divide 135 by 3: \( 135 \div 3 = 45 \)
• Continue dividing by 3: \[ 45 \div 3 = 15 \]

Finally, 5 is a prime number: \( 15 \div 3 = 5 \). So, the prime factorization of 540 is: \[ 540 = 2^2 \times 3^3 \times 5 \].Finding the prime factors is the first step in simplifying radicals.

Square Root Properties

Square roots have some properties that help in their simplification. One important property is that the square root of a product equals the product of the square roots. For instance, \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \].Breaking this down with our example, \sqrt{540} becomes \[ \sqrt{2^2 \times 3^3 \times 5} \].Notice that you can separate the square root of each factor:

• \sqrt{2^2} \rightarrow 2
• \sqrt{3^2} \rightarrow 3
• \sqrt{3 \times 5} = \sqrt{15}

Multiplying, we get \[ 2 \times 3 \times \sqrt{15} = 6 \sqrt{15} \].Using these properties makes complex square roots simpler and easier to manage.

Simplifying Radicals

Simplifying radicals involves expressing the square root in its simplest form. After performing prime factorization and using square root properties, we combine them to simplify the square root fully.Let's use \( \sqrt{540} \) as an example. First step, prime factorize: \[ 540 = 2^2 \times 3^3 \times 5 \]. Next, apply square root properties:

• Group the pairs to simplify: \[ \sqrt{2^2 \times (3^2 \times 3) \times 5} => \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{15} \]= 2 \times 3 \times \sqrt{15}

So, \( \sqrt{540} = 6 \sqrt{15} \).By identifying and then applying these fundamental steps, you simplify any given radical systematically.