Question

One ordered pair (a,b) satisfies the two equations ab^4 = 384 and a^2 b^5 = 4608. What is the value of a in this ordered pair?

Answer 1

Answer: a = 3∛2

Explanation:

ab⁴ = 384 --> a = 384/b⁴

Substitute a = 384/b⁴ into the second equation to solve for "b".

a²b⁵ = 4608

`\bigg((384)/(b^4)\bigg)^2\cdot b^5=4608\\\\\\(147,456b^5)/(b^8)=4608\\\\\\(147,456)/(b^3)=4608\\\\\\(147,456)/(4608)=b^3\\\\\\32=b^3\\\\\\\sqrt[3]{32} =b\\\\\\2\sqrt[3]{4} =b`

Substitute b = 2∛4 into the first equation to solve for "a".

ab⁴ = 384

a(2∛4)⁴ = 384

a = 384/(2∛4)⁴

a = 24/4∛4

= 6/∛4

= 6(∛2)/2

= 3∛2