91Ó°ÊÓ

When we talk about the sixth root of a number, we are looking for a value that, when multiplied by itself six times, equals the original number. In mathematical terms, if you have a number \(x\), the sixth root \((\sqrt[6]{x})\) is a number \(y\) such that \(y^6 = x\).
It's similar to finding a square root or cube root, but here you're taking it to the sixth power. For positive numbers, this is straightforward since numbers like \(64\) have a real sixth root, which would be \(2\) (since \(2^6 = 64\)). However, things get interesting when dealing with negative numbers.
For instance, when considering \(\sqrt[6]{-64}\), we initially look for something that raised to the sixth power gives \(-64\).

• This isn't possible in the real number system without exploring complex numbers.

Real numbers include all the numbers on the number line. This covers everything from fractions, whole numbers, and such, to any number with decimal places. It's a complete continuum. However, real numbers do not include imaginary or complex numbers – that’s where the limitation comes when trying to find roots of negative numbers, like the sixth root of \(-64\).

• If a root of a number exists but isn't a real number, it's usually because it involves taking a root of a negative number, which sends us into the realm of complex numbers.

Real numbers are wonderful for most day-to-day mathematics but sometimes fall short in situations involving roots of negative numbers, leading us to use a broader system.
In the exercise, attempting to find \(\sqrt[6]{-64}\) within the real numbers results in no solution. This is because there’s no real number that can multiply by itself six times to make a negative result.

The complex number system extends the real numbers to include imaginary numbers. Here, **i** represents the square root of \(-1\), which allows us to handle scenarios that real numbers can't, such as the roots of negative numbers.
In the context of the exercise, to find \(\sqrt[6]{-64}\), we split the number into its complex counterparts by using **Euler's formula**.
This involves expressing the root in polar form: \[\sqrt[6]{-64} = 2(cos\left(\frac{\pi + 2\pi k}{6}\right) + i\sin\left(\frac{\pi + 2\pi k}{6}\right))\]

• This formula helps calculate all possible sixth roots when k varies from 0 to 5.

• Real roots aren't possible here; instead, they turn into a principal (main) value and five other unique values derived from the adjustment by angles.

The principal value, when k = 0, results in \(-1 + i\), which represents one of the sixth roots of \(-64\). This demonstrates how complex numbers allow for solutions where real numbers can't provide them.