Question

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

Short answer

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

Step-by-step solution

Factorization

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

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 \( \sqrt{n^2} = n \).

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.

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.