Question

question 3 of 5

select the correct answer.

which expression is in simplest form?

• (2sqrt{63})
• (3sqrt{8})
• (sqrt{132})
• (sqrt{142})

Explanation:

⚡ Using what you learned: simplifying radical expressions

Step 1: Define simplest radical form

A radical expression is in its simplest form if the number under the radical (the radicand) contains no perfect square factors other than \(1\).

Step 2: Analyze each option

• Option 1: \(2\sqrt{63}\)

$$ 63 = 9 \times 7 = 3^2 \times 7 $$

Since \(63\) has a perfect square factor (\(9\)), it is not in simplest form:

$$ 2\sqrt{63} = 2\sqrt{9 \times 7} = 2 \times 3\sqrt{7} = 6\sqrt{7} $$

• Option 2: \(3\sqrt{8}\)

$$ 8 = 4 \times 2 = 2^2 \times 2 $$

Since \(8\) has a perfect square factor (\(4\)), it is not in simplest form:

$$ 3\sqrt{8} = 3\sqrt{4 \times 2} = 3 \times 2\sqrt{2} = 6\sqrt{2} $$

• Option 3: \(\sqrt{132}\)

$$ 132 = 4 \times 33 = 2^2 \times 33 $$

Since \(132\) has a perfect square factor (\(4\)), it is not in simplest form:

$$ \sqrt{132} = \sqrt{4 \times 33} = 2\sqrt{33} $$

• Option 4: \(\sqrt{142}\)

Find the prime factorization of \(142\):

$$ 142 = 2 \times 71 $$

Since both \(2\) and \(71\) are prime numbers, \(142\) has no perfect square factors other than \(1\). Thus, \(\sqrt{142}\) is in simplest form.

Answer:

\(\sqrt{142}\)