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Math  /  Algebra

Question11) Write 163316 \sqrt[3]{3} as an entire radical.

Studdy Solution

STEP 1

1. We are given the expression 163316 \sqrt[3]{3} and need to express it as a single radical.
2. The expression involves a cube root, so we will work with cube roots throughout the process.

STEP 2

1. Express the integer part as a cube root.
2. Combine the cube roots into a single radical expression.

STEP 3

Express the integer 1616 as a cube root. We know that:
16=24 16 = 2^4
To express 1616 as a cube root, we write:
16=(24)=(23)⋅2=233⋅2 16 = (2^4) = (2^3) \cdot 2 = \sqrt[3]{2^3} \cdot 2

STEP 4

Now, express 1616 as a cube root:
16=23⋅23=243 16 = \sqrt[3]{2^3 \cdot 2} = \sqrt[3]{2^4}

STEP 5

Combine the cube roots 243 \sqrt[3]{2^4} and 33 \sqrt[3]{3} into a single radical:
1633=243⋅33 16 \sqrt[3]{3} = \sqrt[3]{2^4} \cdot \sqrt[3]{3}
Using the property of radicals, we can combine them:
=24⋅33 = \sqrt[3]{2^4 \cdot 3}

STEP 6

Simplify the expression inside the cube root:
24⋅3=16⋅3=48 2^4 \cdot 3 = 16 \cdot 3 = 48
So, the entire radical expression is:
483 \sqrt[3]{48}
The expression 163316 \sqrt[3]{3} as an entire radical is:
483 \boxed{\sqrt[3]{48}}

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