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Find the positive solution of the equation. 6x^((6)/(7))+23=24599 Answer:

Find the positive solution of the equation.\newline6x67+23=24599 6 x^{\frac{6}{7}}+23=24599 \newlineAnswer:

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Full solution

Q. Find the positive solution of the equation.\newline6x67+23=24599 6 x^{\frac{6}{7}}+23=24599 \newlineAnswer:
  1. Isolate variable term: First, we need to isolate the term with the variable xx on one side of the equation. To do this, we subtract 2323 from both sides of the equation.\newline6x(6/7)+23−23=24599−236x^{(6/7)} + 23 - 23 = 24599 - 23
  2. Simplify right side: Now, we simplify the right side of the equation by performing the subtraction.\newline6x67=24599−236x^{\frac{6}{7}} = 24599 - 23\newline6x67=245766x^{\frac{6}{7}} = 24576
  3. Divide by 66: Next, we divide both sides of the equation by 66 to solve for x(6/7)x^{(6/7)}. \newline6x(6/7)6=245766\frac{6x^{(6/7)}}{6} = \frac{24576}{6}\newlinex(6/7)=4096x^{(6/7)} = 4096
  4. Recognize power of 22: We recognize that 40964096 is a power of 22. Specifically, 40964096 is 22 raised to the 1212th power (2122^{12}).\newlinex67=212x^{\frac{6}{7}} = 2^{12}
  5. Raise to reciprocal power: To solve for xx, we need to raise both sides of the equation to the reciprocal of 67\frac{6}{7}, which is 76\frac{7}{6}.(x67)76=(212)76(x^{\frac{6}{7}})^{\frac{7}{6}} = (2^{12})^{\frac{7}{6}}
  6. Multiply exponents: When we raise a power to a power, we multiply the exponents. On the left side, (67)×(76)(\frac{6}{7}) \times (\frac{7}{6}) equals 11, so we are left with xx. On the right side, we multiply the exponents 1212 and (76)(\frac{7}{6}).x=212×76x = 2^{12 \times \frac{7}{6}}
  7. Simplify exponent: Now we simplify the exponent on the right side by multiplying 1212 and (7/6)(7/6). \newlinex=284/6x = 2^{84/6}\newlinex=214x = 2^{14}
  8. Calculate final value: Finally, we calculate 2142^{14} to find the value of xx.\newlinex=16384x = 16384

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