A scientist has 50 grams of a radioactive element. The amount of radioactive element remaining after t days can be determined using the equation f of t is equal to 50 times 1 half raised to the t over 5 power After two days the scientist receives a second shipment of 50 grams of the same element. The equation used to represent the amount of shipment 2 remaining after t days is f of t is equal to 50 times 1 half raised to the the fraction with numerator t minus 2 and denominator 5 power Which of the following is an equivalent form of the expression for the amount remaining in shipment 2 ?
Answer
A
the fraction with numerator 50 times 1 half raised to the t over 5 power and denominator 1 half squared
B
the fraction with numerator 50 times 5th root of 1 half to the t th power and denominator 1 half raised to the 2 fifths power
C
the fraction with numerator 50 times 1 half raised to the t minus 2 power and denominator 1 half to the 5th power
D
the fraction with numerator 50 times square root of 1 half to the t th power end root and denominator 1 half to the 5th power
Explanation
Rewrite the exponent as (t − 2)/5 = t/5 − 2/5. Then 50(1/2)^((t−2)/5) = 50(1/2)^(t/5) ÷ (1/2)^(2/5). Since (1/2)^(t/5) is the tth power of the fifth root of 1/2, this matches choice B: 50 times the fifth root of 1/2 to the tth power, divided by (1/2)^(2/5).
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