Question

select the correct answer.

which cube root expression is already in simplified form?

\\(\sqrt3{-1,000}\\)

\\(\sqrt3{-500}\\)

\\(\sqrt3{400}\\)

\\(\sqrt3{900}\\)

Explanation:

⚡ Using what you learned: simplifying radical expressions

Step 1: Define simplified form for cube roots

A cube root expression \(\sqrt[3]{x}\) is in its simplest form if the radicand \(x\) contains no perfect cube factors other than \(1\) or \(-1\).

The smallest positive perfect cubes are:

$$ 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 $$

We need to test each option to see if its radicand is divisible by any of these perfect cubes (other than \(1\)).

Step 2: Analyze each option

• Option 1: \(\sqrt[3]{-1,000}\)

Since \(-1,000 = (-10)^3\), this is a perfect cube and simplifies completely to \(-10\). It is not in simplified radical form.

• Option 2: \(\sqrt[3]{-500}\)

We look for perfect cube factors of \(500\). Since \(500 = 125 \times 4\), and \(125 = 5^3\) is a perfect cube:

$$ \sqrt[3]{-500} = \sqrt[3]{-125 \times 4} = -5\sqrt[3]{4} $$

This can be simplified, so it is not already in simplified form.

• Option 3: \(\sqrt[3]{400}\)

We look for perfect cube factors of \(400\). Since \(400 = 8 \times 50\), and \(8 = 2^3\) is a perfect cube:

$$ \sqrt[3]{400} = \sqrt[3]{8 \times 50} = 2\sqrt[3]{50} $$

This can be simplified, so it is not already in simplified form.

• Option 4: \(\sqrt[3]{900}\)

We find the prime factorization of \(900\):

$$ 900 = 9 \times 100 = 3^2 \times 2^2 \times 5^2 = 2^2 \times 3^2 \times 5^2 $$

None of the prime factors have an exponent of \(3\) or greater. Therefore, \(900\) has no perfect cube factors other than \(1\). This expression cannot be simplified further.

Answer:

\(\sqrt[3]{900}\)