91Ó°ÊÓ

To solve polynomial equations like this one, the first step is to isolate the variable term. This means we move all terms that don’t contain the variable to the other side of the equation. For our example, the equation we start with is:\[ 2x^6 = 128 \]Here, we have to isolate the term involving the variable, which is \( x^6 \). To do this, we divide both sides of the equation by 2:\[ x^6 = \frac{128}{2} \]This simplifies to:\[ x^6 = 64 \]Now, the variable term \( x^6 \) is by itself on one side of the equation. This step of isolating the variable helps set us up for the next step in solving the equation.

Once we have isolated the variable term, we need to solve for the variable itself. In this case, our next step is to take the sixth root of both sides of the equation. Here's what it looks like mathematically:\[ x = \root{6}{64} \]Taking the root is essentially the opposite of taking a power. The sixth root of 64 can also be written as:\[ x = \root{6}{2^6} \]When we're dealing with roots of a number that is also a power, it helps to simplify the expression. Here, we recognize that 64 is the same as \( 2^6 \), making our equation:\[ x = 2 \]Taking roots allows us to simplify exponentiation and isolate the variable directly. It turns a complex power problem into a simple, understandable result.

When dealing with polynomial equations, especially those with even powers, we must remember to consider both the positive and negative roots. In our problem, we arrived at \( x = 2 \) after taking the sixth root of 64. However, we must also consider that:\[ x = -2 \]This is because both positive and negative values, when raised to an even power, result in the same positive number. Hence, for \( x^6 = 64 \), both \( 2^6 \) and \( (-2)^6 \) equal 64. Therefore, the complete solution set for our original equation is:\[ x = 2 \; \text{or} \; x = -2 \]Considering both positive and negative solutions ensures we account for all possible answers, providing a complete and thorough solution to the problem.