Find the value of `root(3)(54872)-root(3)(1728) + root(3)(15625)`

Options

A. 51
B. 49
C. 72
D. 58

Answer

To solve the expression \( \sqrt[3]{54872} - \sqrt[3]{1728} + \sqrt[3]{15625} \), we will break it down step by step.

### Step 1: Calculate \( \sqrt[3]{54872} \)

To find \( \sqrt[3]{54872} \), we can factor it.

- We notice that \( 54872 = 8 \times 6859 \).
- We know that \( 8 = 2^3 \) and \( 6859 = 19^3 \).

Thus, we can write:
\[ \sqrt[3]{54872} = \sqrt[3]{8 \times 6859} = \sqrt[3]{8} \times \sqrt[3]{6859} = 2 \times 19 = 38. \]

### Step 2: Calculate \( \sqrt[3]{1728} \)

Next, we calculate \( \sqrt[3]{1728} \).

- We know that \( 1728 = 12^3 \).

Thus, we have:
\[ \sqrt[3]{1728} = 12. \]

### Step 3: Calculate \( \sqrt[3]{15625} \)

Now, we calculate \( \sqrt[3]{15625} \).

- We know that \( 15625 = 25^3 \).

Thus, we have:
\[ \sqrt[3]{15625} = 25. \]

### Step 4: Substitute the values back into the expression

Now we substitute the values we found back into the original expression:
\[ \sqrt[3]{54872} - \sqrt[3]{1728} + \sqrt[3]{15625} = 38 - 12 + 25. \]

### Step 5: Perform the arithmetic

Now we perform the arithmetic:
\[ 38 - 12 = 26, \]
and then,
\[ 26 + 25 = 51. \]

### Final Answer

Thus, the value of the expression \( \sqrt[3]{54872} - \sqrt[3]{1728} + \sqrt[3]{15625} \) is \( \boxed{51} \).

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