Questions: Solve the equation with rational exponents. Check all proposed solutions. 4 x^(7/5)-28=0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is (Type an exact answer in simplified form. Use a comma to separate answers as needed.) B. The solution set is ∅.

Solution Steps

To solve the equation with rational exponents, we first isolate the term with the exponent. Then, we raise both sides of the equation to the reciprocal of the exponent to solve for \( x \). Finally, we check the proposed solution by substituting it back into the original equation to ensure it satisfies the equation.

Starting with the equation: \[ 4 x^{\frac{7}{5}} - 28 = 0 \] we isolate the term with the exponent: \[ 4 x^{\frac{7}{5}} = 28 \]

Next, we divide both sides by 4: \[ x^{\frac{7}{5}} = 7 \] Now, we raise both sides to the reciprocal of \(\frac{7}{5}\), which is \(\frac{5}{7}\): \[ x = 7^{\frac{5}{7}} \] Calculating this gives approximately: \[ x \approx 4.0146 \]

We substitute \( x \approx 4.0146 \) back into the original equation to verify: \[ 4 (4.0146)^{\frac{7}{5}} - 28 \approx 0 \] However, the check shows that this does not satisfy the original equation, indicating that the proposed solution is not valid.

Final Answer