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Chapter 8: Problem 27

Simplify. $$\sqrt{800}$$

Short Answer

Expert verified
20\sqrt{2}

Step by step solution

01

Prime Factorization

First, find the prime factorization of 800. Break it down into its prime factors: 800 = 2 脳 400 400 = 2 脳 200 200 = 2 脳 100 100 = 2 脳 50 50 = 2 脳 25 25 = 5 脳 5 Thus, 800 = 2^5 脳 5^2
02

Apply the Square Root

To simplify \( \sqrt{800} \), use the prime factorization found in Step 1. Remember that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).\( \sqrt{800} = \sqrt{2^5 \times 5^2} \)
03

Simplify Inside the Square Root

Rewrite the expression to separate the perfect square factors:\( \sqrt{2^5 \times 5^2} = \sqrt{2^4 \times 2 \times 5^2} = \sqrt{(2^2)^2 \times 2 \times 5^2} \).Since \( \sqrt{(2^2)^2} = 2^2 = 4 \) and \( \sqrt{5^2} = 5, \) you get:\( 4 \cdot 5 \cdot \sqrt{2} = 20 \cdot \sqrt{2} \)

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Key Concepts

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

prime factorization
Prime factorization is the process of breaking down a number into its smallest prime factors. Prime factors are numbers that are only divisible by 1 and itself. To start, find two numbers that multiply to give the original number and continue breaking down the factors until all are prime. For example, with the number 800, you start with:
  • 800 = 2 脳 400
  • 400 = 2 脳 200
  • 200 = 2 脳 100
  • 100 = 2 脳 50
  • 50 = 2 脳 25
  • 25 = 5 脳 5
This process results in 800 being expressed as: 800 = 2^5 脳 5^2.
perfect squares
A perfect square is a number that can be expressed as the product of an integer with itself. For example, 25 is a perfect square because it can be written as 5^2. When simplifying square roots, identifying perfect squares within the factorization can greatly simplify the process. In our example, with 800, we have: 800 = 2^5 脳 5^2Since we have both 2^4 (which is a perfect square because 2^4 = (2^2)^2 = 4^2) and 5^2, we can separate these perfect square factors to simplify the square root: 鈭�800 = 鈭�(2^4 脳 2 脳 5^2).
radical expressions
Radical expressions involve roots such as square roots. Simplifying them often requires using prime factorization and identifying perfect squares within the radical. For 鈭�800, we identified the perfect squares in the factorization: 鈭�(2^5 脳 5^2)Separating and taking the square root of perfect squares results in:鈭�(2^4) 脳 鈭�2 脳 鈭�(5^2) = 4 脳 鈭�2 脳 5And finally, simplifying gives us: 20鈭�2. This is a much simpler form of the original square root expression.

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