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Simplify. Assume all variables are positive.\newlinet116×t56t^{\frac{11}{6}} \times t^{\frac{5}{6}}\newlineWrite your answer in the form AA or AB\frac{A}{B}, where AA and BB are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.\newline______

Full solution

Q. Simplify. Assume all variables are positive.\newlinet116×t56t^{\frac{11}{6}} \times t^{\frac{5}{6}}\newlineWrite your answer in the form AA or AB\frac{A}{B}, where AA and BB are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.\newline______
  1. Identify Equation and Apply Property: Identify the equation and apply the property of exponents for multiplication.\newlineWhen multiplying two exponents with the same base, we add the exponents: \newlinetab×tcd=t(ab+cd)t^{\frac{a}{b}} \times t^{\frac{c}{d}} = t^{\left(\frac{a}{b} + \frac{c}{d}\right)}
  2. Add Exponents: Add the exponents (116)(\frac{11}{6}) and (56)(\frac{5}{6}).116+56=11+56\frac{11}{6} + \frac{5}{6} = \frac{11 + 5}{6}
  3. Perform Addition: Perform the addition in the numerator. 11+5=1611 + 5 = 16
  4. Write Result as Exponent: Write the result as a single exponent. t166t^{\frac{16}{6}}
  5. Simplify Fraction: Simplify the fraction 166\frac{16}{6} by dividing both the numerator and the denominator by their greatest common divisor, which is 22. \newline16÷2=816 \div 2 = 8\newline6÷2=36 \div 2 = 3
  6. Write Final Expression: Write the final simplified expression. t83t^{\frac{8}{3}}

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