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Simplify. Assume all variables are positive.\newlinet14t54t^{\frac{1}{4}} \cdot t^{\frac{5}{4}}\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.\newlinet14t54t^{\frac{1}{4}} \cdot t^{\frac{5}{4}}\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 Expression: Identify the expression to be simplified.\newlineWe have the expression t14×t54t^{\frac{1}{4}} \times t^{\frac{5}{4}} and we need to simplify it.
  2. Apply Exponent Property: Apply the property of exponents for multiplying with the same base.\newlineWhen multiplying expressions with the same base, we add the exponents: am×an=a(m+n)a^m \times a^n = a^{(m+n)}.
  3. Add Exponents: Add the exponents of tt. t1/4×t5/4=t1/4+5/4.t^{1/4} \times t^{5/4} = t^{1/4 + 5/4}.
  4. Perform Addition: Perform the addition of the exponents. 14+54=64\frac{1}{4} + \frac{5}{4} = \frac{6}{4}.
  5. Simplify Exponent: Simplify the exponent. 64\frac{6}{4} can be simplified to 32\frac{3}{2} (since 66 divided by 44 equals 1.51.5, which is the same as 32\frac{3}{2}).
  6. Write Final Expression: Write the final simplified expression. t14×t54=t32t^{\frac{1}{4}} \times t^{\frac{5}{4}} = t^{\frac{3}{2}}.

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