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Determine whether each expression is equivalent to 81^(0.5)t+0.25, Equivalent Not Equivalent {:[27^(0.5 t+0.25)*3^(0.5 t+0.25)],[(27^(t+0.5))/(3^(t+0.5))]:} 9^(t)*3

Determine whether each expression is equivalent to 810.5t+0.25 81^{0.5} t+0.25 ,\newlineEquivalent Not Equivalent\newline270.5t+0.25⋅30.5t+0.2527t+0.53t+0.5 \begin{array}{l} 27^{0.5 t+0.25} \cdot 3^{0.5 t+0.25} \\ \frac{27^{t+0.5}}{3^{t+0.5}} \end{array} \newline9t⋅3 9^{t} \cdot 3

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Q. Determine whether each expression is equivalent to 810.5t+0.25 81^{0.5} t+0.25 ,\newlineEquivalent Not Equivalent\newline270.5t+0.25⋅30.5t+0.2527t+0.53t+0.5 \begin{array}{l} 27^{0.5 t+0.25} \cdot 3^{0.5 t+0.25} \\ \frac{27^{t+0.5}}{3^{t+0.5}} \end{array} \newline9t⋅3 9^{t} \cdot 3
  1. Simplify 810.5t+0.2581^{0.5t+0.25}: \newlineStep 11: Simplify 810.5t+0.2581^{0.5t+0.25}.\newline81=3481 = 3^4, so 810.5t+0.25=(34)0.5t+0.25=34(0.5t+0.25)=32t+181^{0.5t+0.25} = (3^4)^{0.5t+0.25} = 3^{4(0.5t+0.25)} = 3^{2t+1}.
  2. Simplify 27(0.5t+0.25)imes3(0.5t+0.25)27^{(0.5t+0.25)} imes 3^{(0.5t+0.25)}: \newlineStep 22: Simplify 27(0.5t+0.25)imes3(0.5t+0.25)27^{(0.5t+0.25)} imes 3^{(0.5t+0.25)}.\newline27=3327 = 3^3, so 27(0.5t+0.25)=(33)(0.5t+0.25)=33imes(0.5t+0.25)=3(1.5t+0.75)27^{(0.5t+0.25)} = (3^3)^{(0.5t+0.25)} = 3^{3 imes (0.5t+0.25)} = 3^{(1.5t+0.75)}.\newline3(0.5t+0.25)3^{(0.5t+0.25)} is already in base 33.\newlineSo, 27(0.5t+0.25)imes3(0.5t+0.25)=3(1.5t+0.75)imes3(0.5t+0.25)=3(1.5t+0.75+0.5t+0.25)=3(2t+1)27^{(0.5t+0.25)} imes 3^{(0.5t+0.25)} = 3^{(1.5t+0.75)} imes 3^{(0.5t+0.25)} = 3^{(1.5t+0.75 + 0.5t+0.25)} = 3^{(2t+1)}.
  3. Simplify 27t+0.53t+0.5 \frac{27^{t+0.5}}{3^{t+0.5}} : Step 33: Simplify 27t+0.53t+0.5 \frac{27^{t+0.5}}{3^{t+0.5}} . 27=33 27 = 3^3 , so 27t+0.5=(33)t+0.5=33(t+0.5)=33t+1.5 27^{t+0.5} = (3^3)^{t+0.5} = 3^{3(t+0.5)} = 3^{3t+1.5} . So, 27t+0.53t+0.5=33t+1.53t+0.5=33t+1.5−t−0.5=32t+1 \frac{27^{t+0.5}}{3^{t+0.5}} = \frac{3^{3t+1.5}}{3^{t+0.5}} = 3^{3t+1.5 - t-0.5} = 3^{2t+1} .
  4. Simplify 9t⋅39^t \cdot 3: \newlineStep 44: Simplify 9t⋅39^t \cdot 3.\newline9=329 = 3^2, so 9t=(32)t=32t9^t = (3^2)^t = 3^{2t}.\newlineSo, 9t⋅3=32t⋅3=32t+19^t \cdot 3 = 3^{2t} \cdot 3 = 3^{2t+1}.

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