Simplify `(root3(8))/(sqrt(16))divsqrt(100)/(sqrt(49))xxroot3(125)`

Options

A. 7
B. `1(3)/(4)`
C. `(7)/(100)`
D. `(4)/(7)`

Answer

To simplify the expression \((\sqrt[3]{8})/(\sqrt{16}) \div (\sqrt{100}/\sqrt{49}) \times \sqrt[3]{125}\), we will follow these steps:

### Step 1: Calculate the cube root of 8
The cube root of 8 is calculated as follows:
\[ \sqrt[3]{8} = 2 \]
**Hint:** Remember that the cube root of a number \(x\) is a value \(y\) such that \(y^3 = x\).

### Step 2: Calculate the square root of 16
The square root of 16 is:
\[ \sqrt{16} = 4 \]
**Hint:** The square root of a number \(x\) is a value \(y\) such that \(y^2 = x\).

### Step 3: Calculate the square root of 100
The square root of 100 is:
\[ \sqrt{100} = 10 \]
**Hint:** Use the same method as in Step 2 to find the square root.

### Step 4: Calculate the square root of 49
The square root of 49 is:
\[ \sqrt{49} = 7 \]
**Hint:** Again, apply the square root definition.

### Step 5: Calculate the cube root of 125
The cube root of 125 is:
\[ \sqrt[3]{125} = 5 \]
**Hint:** Look for a number \(y\) such that \(y^3 = 125\).

### Step 6: Substitute the values back into the expression
Now we can substitute the calculated values back into the expression:
\[ \frac{2}{4} \div \left(\frac{10}{7}\right) \times 5 \]

### Step 7: Simplify \(\frac{2}{4}\)
\[ \frac{2}{4} = \frac{1}{2} \]
**Hint:** Simplify fractions by dividing the numerator and denominator by their greatest common divisor.

### Step 8: Rewrite the division as multiplication
The division can be rewritten as multiplication by the reciprocal:
\[ \frac{1}{2} \times \frac{7}{10} \times 5 \]

### Step 9: Multiply the fractions
Now we can multiply:
\[ \frac{1 \times 7 \times 5}{2 \times 10} = \frac{35}{20} \]
**Hint:** When multiplying fractions, multiply the numerators together and the denominators together.

### Step 10: Simplify \(\frac{35}{20}\)
\[ \frac{35}{20} = \frac{7}{4} \]
**Hint:** Again, simplify by finding the greatest common divisor.

### Final Answer
Thus, the simplified form of the expression is:
\[ \frac{7}{4} \]