A biologist begins working with a sample containing 20 comma 000 bacteria. The population size doubles every 10 days according to the equation p times t is equal to 20 comma 000 times 2 raised to the t over 10 power , where t represents the number of days. After seven days the biologist begins working with a second sample of 20 comma 000 bacteria. The equation used to represent the population of sample 2 after t days is p times t is equal to 20 comma 000 times 2 raised to the the fraction with numerator t minus 7 and denominator 10 power Which of the following is an equivalent form of the equation for the population of sample 2 ?
Answer
A
p times t is equal to the fraction with numerator 20 comma 000 times 2 raised to the t minus 7 power and denominator 2 to the 10th power
B
p times t is equal to the fraction with numerator 20 comma 000 10th root of 2 to the t th power and denominator 10th root of 2 to the 7th power
C
p times t is equal to the fraction with numerator 20 comma 000 times 7th root of 2 to the t th power and denominator 2 to the 10th power
D
p times t is equal to the fraction with numerator 20 comma 000 times 2 raised to the t over 10 power and denominator 2 to the 7th power
Explanation
Rewrite the exponent as (t − 7)/10 = t/10 − 7/10. Then use the quotient rule for exponents: 2^(t/10) ÷ 2^(7/10). Since 2^(1/10) is the 10th root of 2, this becomes (10th root of 2)^t divided by (10th root of 2)^7, so choice B is equivalent.
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