Problem 78 - Algebra 1

Question

Simplify the radical expression. $$ \sqrt{200} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{200} \) is \( 10\sqrt{2} \).

Step 1: Factorization

To simplify \( \sqrt{200} \), begin by factoring 200. The prime factorization of 200 is \( 2^3 \times 5^2 \).

Step 2: Identifying the Pairs

Now look for pairs of factors. In the prime factorization of 200, there is one pair of 2’s and one pair of 5’s. Pairing terms under a square root allows us to simplify because \( \sqrt{n^2} = n \).

Step 3: Simplifying the Pairs

We can pull these pairs out from under the root. For each pair of Twos, a Two can come out. Hence, one 2 will come out of the radical. For each pair of Fives, a Five can come out. Hence, one 5 will come out of the radical.

Step 4: Final Simplification

Multiply the values obtained in Step 3 to get the simplified form of \( \sqrt{200} \). In this case, 2 and 5 multiply to 10. There is still a 2 under the root that has no pair, so \( \sqrt{2} \) is the final part of the simplified form.

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers. When simplifying radicals, prime factorization is a crucial first step, because it allows us to find perfect squares within a number.
For instance, with the number 200, we need to break it down into its prime components. Begin by dividing by the smallest prime number, which is 2. After dividing by 2 three times, we end up with a quotient of 25. Then, notice that 25 is divisible by 5, another prime number, resulting in the factorization of 200 as \( 2^3 \times 5^2 \).
By listing the number's factors as prime numbers, you can easily identify and pair them up. Remember, a prime factor is an integer greater than 1 that has no other divisors besides 1 and itself.

Square Roots

The operation of taking the square root involves finding a number which, when multiplied by itself, gives the original number. When you see a radical symbol \( \sqrt{} \), it signifies you should perform this operation.
In the context of simplifying expressions like \( \sqrt{200} \), recognizing and simplifying square roots becomes essential. Having established the prime factorization \( 2^3 \times 5^2 \), we can simplify the root by looking for pairs. Each pair corresponds to a perfect square, which can then be pulled out from under the radical.
The presence of a pair gives an opportunity to simplify since \( \sqrt{n^2} = n \). In our example, each group of \( 2^2 \) or \( 5^2 \) simplifies because they create whole numbers: \( \sqrt{4} = 2 \) and \( \sqrt{25} = 5 \). Concisely, this makes double checking your prime factorization with pairs important for successful simplification.

Simplifying Expressions

Simplifying expressions generally means reducing them to their simplest form, making them easier to understand and work with. When working with radicals like \( \sqrt{200} \), simplification involves using steps such as prime factorization and leveraging the properties of square roots.
Following the identification of pairs in the prime factorized form \( 2^3 \times 5^2 \), we separate the number into parts that can come outside the square root and those that remain inside. For example, from \( 200 = 2^3 \times 5^2 \), each pair of 2’s (\( 2^2 \)) and 5’s (\( 5^2 \)) can bring one 2 and one 5 out of the radical.
Consequently, the simplified form becomes \( 2 \times 5 \times \sqrt{2} = 10\sqrt{2} \). The important takeaway in simplifying, especially with radicals, is ensuring all possible pairs have been removed to make the expression as clean and straightforward as possible.