Question

question
find the positive solution of the equation.
$5x^{\frac{7}{3}} - 18 = 10485742$

Explanation:

Step1: Isolate the variable term

Add 18 to both sides.
$$5x^{\frac{7}{3}} = 10485742 + 18$$
$$5x^{\frac{7}{3}} = 10485760$$

Step2: Solve for $x^{\frac{7}{3}}$

Divide both sides by 5.
$$x^{\frac{7}{3}} = \frac{10485760}{5}$$
$$x^{\frac{7}{3}} = 2097152$$

Step3: Eliminate the fractional exponent

Raise both sides to the $\frac{3}{7}$ power.
$$x = (2097152)^{\frac{3}{7}}$$

Step4: Simplify the base

Note that $2097152 = 2^{21}$.
$$x = (2^{21})^{\frac{3}{7}}$$

Step5: Simplify the exponent

Multiply the exponents.
$$x = 2^{21 \times \frac{3}{7}}$$
$$x = 2^9$$

Step6: Calculate the final value

Compute $2^9$.
$$x = 512$$

Answer:

$512$