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Find the positive solution of the equation. 7x^((3)/(7))+19=1531 Answer:

Find the positive solution of the equation.\newline7x37+19=1531 7 x^{\frac{3}{7}}+19=1531 \newlineAnswer:

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Full solution

Q. Find the positive solution of the equation.\newline7x37+19=1531 7 x^{\frac{3}{7}}+19=1531 \newlineAnswer:
  1. Subtract 1919: Subtract 1919 from both sides of the equation to isolate the term with the variable.\newline7x37+19−19=1531−197x^{\frac{3}{7}} + 19 - 19 = 1531 - 19
  2. Simplify equation: Perform the subtraction to simplify the equation.\newline7x37=15127x^{\frac{3}{7}} = 1512
  3. Divide by 77: Divide both sides of the equation by 77 to solve for x37x^{\frac{3}{7}}.\newline7x377=15127\frac{7x^{\frac{3}{7}}}{7} = \frac{1512}{7}
  4. Recognize perfect cube: Perform the division to simplify the equation.\newlinex37=216x^{\frac{3}{7}} = 216
  5. Raise to power: Recognize that 216216 is a perfect cube, as 63=2166^3 = 216.
  6. Simplify left side: Raise both sides of the equation to the power of 73\frac{7}{3} to solve for x.\newline(x37)73=21673(x^{\frac{3}{7}})^{\frac{7}{3}} = 216^{\frac{7}{3}}
  7. Simplify right side: Simplify the left side of the equation by multiplying the exponents.\newlinex37⋅73=x1=xx^{\frac{3}{7} \cdot \frac{7}{3}} = x^1 = x
  8. Calculate value: Simplify the right side of the equation by finding the 77th power of 66 and then taking the cube root.\newline21673=(63)73=63⋅73=67216^{\frac{7}{3}} = (6^3)^{\frac{7}{3}} = 6^{3 \cdot \frac{7}{3}} = 6^7
  9. Conclude positive solution: Calculate 676^7 to find the value of x.\newline67=2799366^7 = 279936
  10. Conclude positive solution: Calculate 676^7 to find the value of x.\newline67=2799366^7 = 279936Conclude that the positive solution of the equation is x=279936x = 279936.

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