Problem 60
Question
Simplify. $$ \sqrt{252} $$
Step-by-Step Solution
Verified Answer
The simplified form of \( \sqrt{252} \) is \( 6 \sqrt{7} \).
1Step 1: Prime Factorization
Break down 252 into its prime factors. Start by dividing by the smallest prime number (2) and continue dividing by prime numbers.\[ 252 \ \frac{252}{2} = 126 \ \frac{126}{2} = 63 \ \frac{63}{3} = 21 \ \frac{21}{3} = 7 \ \frac{7}{7} = 1 \]So, the prime factorization of 252 is \(2^2 \cdot 3^2 \cdot 7\).
2Step 2: Group Factors into Pairs
Identify pairs of prime factors to simplify the square root. Each pair of identical prime factors can be reduced to a single factor outside the square root.\[ 2^2 \cdot 3^2 \cdot 7 = (2 \times 2) \cdot (3 \times 3) \cdot 7 \]
3Step 3: Simplify the Square Root
Since each pair of factors can be taken out of the square root as a single factor, we get:\[ \sqrt{2^2 \cdot 3^2 \cdot 7} = 2 \cdot 3 \cdot \sqrt{7} \]So, the simplified form of \( \sqrt{252} \) is \( 6 \sqrt{7} \).
Key Concepts
Prime FactorizationSimplifying RadicalsPairing Factors
Prime Factorization
Prime factorization is the process of breaking down a composite number into its prime factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. To perform prime factorization, you start by dividing the number by the smallest prime number (which is 2) and continue dividing by prime numbers until you are left with 1.
For example, to find the prime factors of 252:
For example, to find the prime factors of 252:
- 252 divided by 2 gives 126
- 126 divided by 2 gives 63
- 63 divided by 3 gives 21
- 21 divided by 3 gives 7
- Finally, 7 divided by 7 gives 1
Simplifying Radicals
Simplifying radicals involves expressing the number or expression inside the square root in its simplest form. This often involves prime factorization and recognizing pairs of prime factors.
For instance, with \(\sqrt{252}\), we already know its prime factors are \(2^2 \cdot 3^2 \cdot 7\). Simplifying means we should identify pairs of the same number within these factors, because each pair can be taken out of the square root as a single number.
By recognizing and simplifying these pairs: \(2^2\) and \(3^2\), we can take them out of the square root one at a time. This results in: \( 2 \times 3 \times \sqrt{7} \), which simplifies to \(6\sqrt{7}\).
For instance, with \(\sqrt{252}\), we already know its prime factors are \(2^2 \cdot 3^2 \cdot 7\). Simplifying means we should identify pairs of the same number within these factors, because each pair can be taken out of the square root as a single number.
By recognizing and simplifying these pairs: \(2^2\) and \(3^2\), we can take them out of the square root one at a time. This results in: \( 2 \times 3 \times \sqrt{7} \), which simplifies to \(6\sqrt{7}\).
Pairing Factors
Pairing factors is a crucial step in simplifying radicals, especially square roots. When you have prime factored a number, you look for pairs of the same prime number. Each pair can be taken out of the square root as a single factor.
Let’s consider our example: \(2 \cdot 2, 3 \cdot 3, and 7\). Here, \(2^2\) and \(3^2\) each become a single 2, and a single 3 respectively when they come out of the square root, while the 7 remains inside the square root since it doesn’t form a pair.
Hence, \sqrt{2^2 \cdot 3^2 \cdot 7}\ becomes \[ 2 \cdot 3 \cdot \sqrt{7} = 6 \sqrt{7} \].
This process of pairing factors transforms a complex expression into a much simpler one you can easily understand and work with.
Let’s consider our example: \(2 \cdot 2, 3 \cdot 3, and 7\). Here, \(2^2\) and \(3^2\) each become a single 2, and a single 3 respectively when they come out of the square root, while the 7 remains inside the square root since it doesn’t form a pair.
Hence, \sqrt{2^2 \cdot 3^2 \cdot 7}\ becomes \[ 2 \cdot 3 \cdot \sqrt{7} = 6 \sqrt{7} \].
This process of pairing factors transforms a complex expression into a much simpler one you can easily understand and work with.