Question

1. write \\(\sqrt{1694}\\) in simplest form.

Explanation:

Find prime factors of the radicand

We begin by finding the prime factorization of \(1694\).
Since \(1694\) is even, we divide by \(2\):

$$1694 \div 2 = 847$$

Now we find factors of \(847\). Testing small primes:

• It is not divisible by \(3\) or \(5\).
• Testing \(7\): \(847 \div 7 = 121\).

Thus, we have:

$$1694 = 2 \times 7 \times 121$$

Identify perfect square factors

Using the Radical Simplification knowledge point, we look for perfect squares.
We recognize that \(121\) is a perfect square:

$$121 = 11^2$$

So, the prime factorization of \(1694\) is:

$$1694 = 2 \times 7 \times 11^2 = 14 \times 11^2$$

Simplify the radical expression

We rewrite the radical using the product property of square roots:

$$\sqrt{1694} = \sqrt{11^2 \times 14}$$

We separate the perfect square from the remaining factors:

$$\sqrt{11^2 \times 14} = \sqrt{11^2} \times \sqrt{14}$$

Simplifying the perfect square yields:

$$11\sqrt{14}$$

Answer:

\(11\sqrt{14}\)