Express the following in the simplest form: `root(3)(625)`

Answer

To express the cube root of 625 in its simplest form, we can follow these steps:

### Step 1: Prime Factorization of 625
First, we need to find the prime factors of 625.

- Divide 625 by 5:
\[ 625 \div 5 = 125 \]
- Divide 125 by 5:
\[ 125 \div 5 = 25 \]
- Divide 25 by 5:
\[ 25 \div 5 = 5 \]
- Finally, divide 5 by 5:
\[ 5 \div 5 = 1 \]

So, the prime factorization of 625 is:
\[ 625 = 5 \times 5 \times 5 \times 5 = 5^4 \]

### Step 2: Rewrite the Cube Root
Now we can rewrite the cube root of 625 using its prime factorization:
\[ \sqrt[3]{625} = \sqrt[3]{5^4} \]

### Step 3: Simplify the Cube Root
We can separate the powers of 5:
\[ \sqrt[3]{5^4} = \sqrt[3]{5^3 \times 5^1} = \sqrt[3]{5^3} \times \sqrt[3]{5^1} \]

### Step 4: Evaluate the Cube Root
The cube root of \(5^3\) is 5:
\[ \sqrt[3]{5^3} = 5 \]
Thus, we have:
\[ \sqrt[3]{5^4} = 5 \times \sqrt[3]{5} \]

### Final Answer
Therefore, the simplest form of \(\sqrt[3]{625}\) is:
\[ 5 \times \sqrt[3]{5} \]

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