Write all of the values of 𝑥 which satisfy the equation two 𝑥 to the fourth power plus eight equals 13. If necessary, give your answer or answers to four significant figures.
In this equation, to solve for 𝑥, we’re going to start by rearranging it so that we just have the 𝑥 term on its own on one side of the equation. Beginning by subtracting eight from both sides of the equation, we’ll have two 𝑥 to the fourth power equals 13 minus eight, which can be simplified to two 𝑥 to the fourth power equals five. On the left-hand side, we have two multiplied by 𝑥 to the fourth power, which means that we need to do the inverse operation, which is dividing by two, which gives us 𝑥 to the fourth power equals five over two.
Let’s pause for a second and recall what it means to have a number taken to the fourth power. If we have 𝑎 to the fourth power, this can be written as 𝑎 times 𝑎 times 𝑎 times 𝑎. So if we had two to the fourth power, this is equivalent to two times two times two times two, which can be evaluated as 16 since two times two is four. And the other two times two is four. And multiplying those would give us 16.
Equally, if we had negative two to the fourth power, that would be negative two times negative two times negative two times negative two, which is also equal to 16 since our two lots of negative values multiplied give us positive values. So then, if we look at our 16 values and we want to take the fourth root, we know that there will be two possible answers, two or negative two. Therefore, when we come to the next stage of our rearranging and we take the fourth root, we know that there will be two answers, a positive value and a negative value.
So we have 𝑥 equals the fourth root of five over two. Using our calculator, we can evaluate this as 1.25743343 and so on. We take both the positive and negative root values and can indicate this with the plus negative symbol in front. Rounding our answers to four significant figures here means that we need to check the fourth decimal digit. As it is not equal to five or more, then our answer remains as plus or minus 1.257.
And therefore, our final answers for the values of 𝑥 are the positive and negative values — 1.257, negative 1.257.