Simplifying the Radical Expression

The problem asks to evaluate the expression: $ \frac{\sqrt[3]{6859}}{\sqrt[4]{256}} \times \frac{2}{57} \times 168 $ To solve this, we first find the values of the cube root and the fourth root.

Calculating Roots

• Cube Root Calculation: We need to find a number whose cube is 6859. Testing integer values, we find that $19^3 = 19 \times 19 \times 19 = 6859$. Therefore, $\sqrt[3]{6859} = 19$.
• Fourth Root Calculation: We need to find a number whose fourth power is 256. We know that $4^4 = 4 \times 4 \times 4 \times 4 = 256$. Therefore, $\sqrt[4]{256} = 4$.

Substituting and Simplifying

Substitute the calculated root values back into the expression:

Now, simplify the expression step-by-step:

1. Combine fractions: Notice that $57 = 19 \times 3$. Simplify the first two terms: $ \frac{19}{4} \times \frac{2}{57} = \frac{19}{4} \times \frac{2}{19 \times 3} $ Cancel out the common factor 19: $ \frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6} $
2. Multiply by the remaining term: Now multiply the result by 168: $ \frac{1}{6} \times 168 $ Perform the division: $ \frac{168}{6} = 28 $

The value of the expression is 28.