Question

solving equations with fractional exponents
score: 5/10 penalty: none
question
find the positive solution of the equation.
$4x^{\frac{7}{6}} + 16 = 19131892$

Explanation:

Step1: Isolate the variable term

Subtract 16 from both sides:
$4x^{\frac{7}{6}} = 19131892 - 16$
$4x^{\frac{7}{6}} = 19131876$

Step2: Solve for $x^{\frac{7}{6}}$

Divide both sides by 4:
$x^{\frac{7}{6}} = \frac{19131876}{4}$
$x^{\frac{7}{6}} = 4782969$

Step3: Eliminate the fractional exponent

Raise both sides to the power of $\frac{6}{7}$:
$x = 4782969^{\frac{6}{7}}$
Note that $4782969 = 9^7$, so substitute:
$x = (9^7)^{\frac{6}{7}}$
$x = 9^{7 \times \frac{6}{7}}$
$x = 9^6$

Step4: Calculate $9^6$

$9^6 = 9 \times 9 \times 9 \times 9 \times 9 \times 9$
$9^2=81$, $9^3=729$, $9^6=(9^3)^2=729^2=531441$

Answer:

$531441$