Oxford University Entrance Exam Question | Can you solve ?

Added: 2026-05-06

[00:00:01]Hello, welcome back once again. Today we're going to solve this strange looking equation. The -8 to the^ of n is equal to 18.

[00:00:11]Now take note as far as we have a negative base here and we have to solve for a real number n. There's no how a negative number raised to the power of some value will give us the positive of that base. Right? So this tells us that x n I mean cannot be a real number right. Therefore we go with complex solutions or we go for complex solutions. So let's get started. Now let's take note of the following -1 is equal to according to lonil e ra to the power of i. So generally -1 is equal to e rais^ of i into bracket 2 k + 1. So here k can be any integer.

[00:01:03]Now from our problem from the base here we have -1 * 18. Then rest of n this is equal to 18.

[00:01:14]Now let us take note of the following.

[00:01:19]A can be written in the form e ra to the^ of ln of a. So we're going to apply that property. So from this base we have e to the^ of i pi into brackets 2 k + 1 then multip by 18. Use this property it becomes e^ ln of 18. Then rais the^ of n. This is equal to e rais^ of ln of 18.

[00:01:49]Now we have product and we have the same base. We're going to add the exponents.

[00:01:54]So here we get e to the^ of i pi into bracket 2 k + 1 then plus ln of 18 then rais^ of n this is equal to e^ of ln of 18. So here these powers multiplies together right. So we get e^ n into bracket i into bracket 2 k + 1 + ln of 18.

[00:02:26]So this is equal to e rais^ ln of 18.

[00:02:30]Now we have the same base from both sides. So we're going to equate the exponents. So this implies from here that n into bracket i pi into bracket 2k + 1 + ln of 18 is equal to ln of 18. Now divide both sides of the equation by the coefficient of x which is this. So we get here I mean coefficient of n. So we get n is equal to ln of 18 divided by i into bracket 2 k + 1 then plus ln of 18 here k is an integer. So here this give us the general solution for n and you can see for k to be an integer. So it shows that the solution here is infinite and this is the solution to the problem.

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