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Chapter 10: Problem 25

Simplify completely. If the radical is already simplified, then say so. $$\sqrt{400}$$

Short Answer

Expert verified
The square root of 400 is already simplified and is equal to 20.

Step by step solution

01

Find the prime factorization of 400

To find the prime factorization of 400, divide 400 by the smallest prime number (2) repeatedly until the quotient is a prime number or cannot be divided further. 400 梅 2 = 200 200 梅 2 = 100 100 梅 2 = 50 50 梅 2 = 25 The quotient (25) is not a prime number, but it cannot be divided by 2 further, so we move to the next smallest prime number (3). Since 25 is not divisible by 3, we move to the next smallest prime number (5). 25 梅 5 = 5 The quotient (5) is a prime number, so we are done with the prime factorization. Therefore, the prime factorization of 400 is: \(400 = 2^4 * 5^2\)
02

Simplify the square root

Now that we have the prime factorization, we can simplify the square root. For each pair of the same factor inside the square root, one factor can be taken out of the root. \(\sqrt{400} = \sqrt{2^4 * 5^2}\) Since there are two pairs of 2s and two pairs of 5s, we can take one 2 and one 5 out of the square root, which gives: \(\sqrt{400} = 2 * 2 * 5 * 5 = 20\) The square root of 400 is 20.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factorization
Prime factorization breaks a number down into its fundamental building blocks鈥攑rime numbers. A prime number is one with only two distinct positive divisors: 1 and itself. This means it cannot be divided evenly by any other numbers. To find the prime factors of 400:
  • Start with the smallest prime, 2. Divide repeatedly until it no longer divides evenly.
  • 400 梅 2 = 200, so we write down one 2 and continue with 200.
  • 200 梅 2 = 100, add another 2 and continue with 100.
  • 100 梅 2 = 50, add another 2 and continue with 50.
  • 50 梅 2 = 25, add another 2 and continue with 25.
  • 25 isn't divisible by 2. Next prime is 3, but 25 isn't divisible by 3 either.
  • Move to 5, the next prime: 25 梅 5 = 5, add a 5; 5 梅 5 = 1, add another 5.
Now, all factors are prime, with 400 expressed as the product of primes: \[400 = 2^4 \times 5^2\] This factorization will be crucial for further simplification steps.
Simplifying Radicals
Simplifying radicals means expressing the root in its simplest form. Radicals often contain numbers that can be reduced or pulled out from under the square root sign. Simplifying involves using prime factorization results:
  • Start with \(\sqrt{400}\), and Gnow use 400's prime factorization: \(2^4 \times 5^2\).
  • Identify perfect squares formed inside the radical. Here, two pairs of 2s (\(2^4\)) and a pair of 5s (\(5^2\)) make perfect squares.
  • For each pair of identical numbers in the prime factorization, take one number out of the radical.
  • From \(2^4\), you can take 2 out twice, and from \(5^2\), take 5 out once.
Thus, \(\sqrt{2^4 \times 5^2} = 2 \times 2 \times 5 = 20\). This neat expression means the radical is fully simplified.
Perfect Squares
Perfect squares are numbers that can be expressed as the product of an integer with itself. In radical simplification, they are the key to breaking down complex roots. Understanding perfect squares is crucial when working with square roots:
  • They simplify expressions since knowing perfect squares helps in factorizing numbers under the root.
  • In \(400\), the perfect squares in \(2^4\) and \(5^2\) simplify the task.
  • \(2^2 = 4\) and \(5^2 = 25\), both are perfect squares, and their whole number result is 2 and 5, respectively, when rooted.
Thus, leveraging perfect squares, we find: \(\sqrt{400}\) simplifies to \(20\) because \(\sqrt{4} = 2\) and \(\sqrt{25} = 5\). Perfect squares ensure a streamlined route to the simplest expression of radicals.

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