A community library's server storage increases by a factor of \(3^{4} \div 3^{-2}\) gigabytes after a first upgrade, and then by a further factor of \(3^{-1...
Question 1 Report
A community library's server storage increases by a factor of \(3^{4} \div 3^{-2}\) gigabytes after a first upgrade, and then by a further factor of \(3^{-1}\) once a second upgrade is applied.
- Work out \(3^{4} \div 3^{-2}\) as a single power of 3, and evaluate it. (2)
- Work out the total storage after the second upgrade is applied, giving your answer as an integer. (2)
- Express your answer to part (b) as a power of 3. (1)
Answer Details
The laws of indices \(a^{m}\div a^{n} = a^{m-n}\) and \(a^{m}\times a^{n}=a^{m+n}\) apply equally when one of the exponents is negative, provided the sign is handled carefully.
(a) Dividing powers of the same base subtracts the exponents: \(3^{4}\div3^{-2} = 3^{4-(-2)} = 3^{6} = 729\) gigabytes. [2 marks]
(b) Multiplying by the second factor: \(729\times3^{-1} = 729\div3 = 243\) gigabytes. [2 marks]
(c) Since \(3^{5}=243\), the total storage is \(3^{5}\) gigabytes. [1 mark]
Exam tip: subtracting a negative exponent, as in \(4-(-2)\), always increases the result - check the sign carefully rather than treating it as \(4-2\).
Download The App On Google Playstore
Everything you need to excel in your exams
