Question

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what is the simplified form of this expression?

\\\sqrt5{-1,280}\\

Explanation:

⚡ Using what you learned: Simplifying Radical Expressions

Step 1: Factor out the negative sign

Since the index of the radical is odd (a cube root), we can pull the negative sign out of the radical:

$$ \sqrt[3]{-1,280} = -\sqrt[3]{1,280} $$

Step 2: Find the largest perfect cube factor

Find the prime factorization of \( 1,280 \) to identify perfect cubes:

$$ 1,280 = 128 \times 10 = 2^7 \times 2 \times 5 = 2^8 \times 5 $$

Group the factors into perfect cubes:

$$ 1,280 = 2^6 \times 2^2 \times 5 = 64 \times 4 \times 5 = 512 \times 2.5 \text{ (not integer)} $$

Alternatively, check perfect cubes directly:

$$ 8^3 = 512 $$

$$ 1,280 = 512 \times 2.5 \text{ (not an integer)} $$

$$ 4^3 = 64 $$

$$ 1,280 = 64 \times 20 $$

So, the largest perfect cube factor of \( 1,280 \) is \( 64 \):

$$ 1,280 = 64 \times 20 $$

Step 3: Simplify the radical

Apply the product property of radicals:

$$ -\sqrt[3]{1,280} = -\sqrt[3]{64 \times 20} $$

$$ -\sqrt[3]{64} \times \sqrt[3]{20} = -4\sqrt[3]{20} $$

Answer:

$$ -4\sqrt[3]{20} $$