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Sample Question 1:
Step 1: Take note of the given expression \(\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}\).
Step 2: Simplify each square root in the expression by adding up the numbers under each square root: this gives \(\sqrt{1} + \sqrt{4} + \sqrt{9} + \sqrt{16}\).
Step 3: Now, calculate the square root of each number: this results in \(1 + 2 + 3 + 4\).
Step 4: Add those numbers together for a final result of \(10\).
Step 5: So, the value in simplest form of the given expression is \(\boxed{\textbf{(C)} 10}\).
Note: The pattern observed here is that the sum of the first 'n' odd numbers is \(n^2\). This can be helpful in quickly simplifying expressions in a similar format.
What is the value in simplest form of the following expression?\(\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}\)
\(\textbf{(A) }5 \qquad \textbf{(B) }4 + \sqrt{7} + \sqrt{10} \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 15 \qquad \textbf{(E) } 4 + 3\sqrt{3} + 2\sqrt{5} + \sqrt{7}\)
Answer Keys
Question 1: CSolutions
Question 1Step 1: Take note of the given expression \(\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}\).
Step 2: Simplify each square root in the expression by adding up the numbers under each square root: this gives \(\sqrt{1} + \sqrt{4} + \sqrt{9} + \sqrt{16}\).
Step 3: Now, calculate the square root of each number: this results in \(1 + 2 + 3 + 4\).
Step 4: Add those numbers together for a final result of \(10\).
Step 5: So, the value in simplest form of the given expression is \(\boxed{\textbf{(C)} 10}\).
Note: The pattern observed here is that the sum of the first 'n' odd numbers is \(n^2\). This can be helpful in quickly simplifying expressions in a similar format.