Cube root of 81 simplified

Understand the Problem

The question is asking for the simplified form of the cube root of 81. To solve this, we will express 81 in terms of its prime factors and find the cube root.

Answer

$3 \sqrt[3]{3}$

Steps to Solve

Prime factorization of 81

First, we need to break down 81 into its prime factors.

$$ 81 = 3 \times 3 \times 3 \times 3 = 3^4 $$

Using the properties of cube roots

Now we can apply the property of cube roots that states:

$$ \sqrt[3]{a^m} = a^{m/3} $$

In this case, we can express the cube root of (81) as:

$$ \sqrt[3]{81} = \sqrt[3]{3^4} $$

Simplification of the cube root

Now we can simplify the expression:

$$ \sqrt[3]{3^4} = 3^{4/3} = 3^{1 + 1/3} = 3^1 \times 3^{1/3} = 3 \sqrt[3]{3} $$

Thus, the simplified form of the cube root of 81 is:

$$ 3 \sqrt[3]{3} $$

The simplified form of the cube root of 81 is $3 \sqrt[3]{3}$.

More Information

The cube root simplifies the expression by breaking it down into prime factors. Understanding the relationship between exponents and roots is helpful for simplifying similar radical expressions.

Tips

Forgetting to factor the number properly into its prime components can lead to incorrect cube roots.
Confusing cube roots with square roots leading to arithmetic errors.

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