Multiply : `root(3)(5)` by `root(4)(2)`.

Answer

To solve the problem of multiplying \( \sqrt[3]{5} \) by \( \sqrt[4]{2} \), we can follow these steps:

### Step 1: Rewrite the roots in exponential form
We can express the roots as powers:
\[ \sqrt[3]{5} = 5^{1/3} \quad \text{and} \quad \sqrt[4]{2} = 2^{1/4} \]

### Step 2: Multiply the two expressions
Now we multiply the two expressions:
\[ 5^{1/3} \times 2^{1/4} \]

### Step 3: Find a common denominator for the exponents
To combine these two terms, we need a common denominator for the exponents. The denominators are 3 and 4. The least common multiple (LCM) of 3 and 4 is 12.

### Step 4: Rewrite the exponents with the common denominator
We can rewrite the exponents with the common denominator:
\[ 5^{1/3} = 5^{4/12} \quad \text{and} \quad 2^{1/4} = 2^{3/12} \]

### Step 5: Combine the bases
Now we can combine the bases:
\[ 5^{4/12} \times 2^{3/12} = (5^4 \times 2^3)^{1/12} \]

### Step 6: Calculate the powers
Now we calculate \( 5^4 \) and \( 2^3 \):
\[ 5^4 = 625 \quad \text{and} \quad 2^3 = 8 \]

### Step 7: Multiply the results
Now we multiply these results:
\[ 625 \times 8 = 5000 \]

### Step 8: Write the final result
Thus, we have:
\[ \sqrt[12]{5000} \]

### Final Answer:
The final answer is:
\[ \sqrt[12]{5000} \]

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