# Video: Solving Equations Involving Fractional Indices

Given that 3𝑥^(3/2) − 4 = 1, find the value of 𝑥. If necessary, give your answer to 3 decimal places.

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### Video Transcript

Given that three 𝑥 to the power of three over two minus four equals one, find the value of 𝑥. If necessary, give your answer to three decimal places.

So, the first thing we want to do with this equation is have our 𝑥 term on its own on the left-hand side. So what we’re going to do is we’re gonna add four to each side of the equation. And when we add four to each side of the equation, what we’re gonna be left with is three 𝑥 to the power of three over two equals five. And then, what we’re gonna do is divide each side of the equation by three. So, we get 𝑥 to the power of three over two is equal to five over three or five-thirds.

So, now, in order to solve the problem and find the value of 𝑥, what we need to do is have a look at the exponent. So, we’ve got 𝑥 to the power of three over two. Well, what does this mean? Well, we can have a quick look at an exponent rule. Well, if we have 𝑥 to the power of 𝑎 over 𝑏, this is gonna be equal to the 𝑏th root of 𝑥, to the power of 𝑎, or the 𝑏th root of 𝑥 to the power of 𝑎.

So, we can actually see that you could write it either way cause we can either deal with the root first or the exponent first. So, what we’re gonna do first is we’re gonna rewrite 𝑥 to the power of three over two. And we can rewrite this as the square root of 𝑥 cubed. And we’ve chosen to do it this way just because this is probably easier to start off with. So, now, to use inverse operations, the first thing we’re gonna do is square each side of the equation. And it’s worth noting that if we square a fraction, so, for instance, if we have 𝑎 over 𝑏 all squared, then this is the same as 𝑎 squared over 𝑏 squared. So, you square the numerator and square the denominator.

Well, in that case, if we square five, we get 25. And if we square three, we get nine. So, we get now 𝑥 cubed is equal to 25 over nine. So then, what we need to do is take the cube root of both sides of the equation cause, again, we’re gonna use the inverse operation. So, this’s gonna leave us with 𝑥 is equal to the cube root of 25 over nine. Well, if we type this into our calculator, what we’re gonna get is 𝑥 is equal to 1.405721109.

But have we finished here? Well, no, we need to check what accuracy the question wants us to leave our answer in. Well, the question says it wants our answer left to three decimal places. Well, if we take a look at the third decimal place, this is where we’ve got five. And then, the number to the right of this is our deciding number. And we can see that this is a seven. And because it’s a seven, it means that it’s greater than five or five and greater. And if it’s five and greater, then what we do is we round up our number to the left of this digit. So, when we do this, we get 𝑥 is equal to 1.406. And this is, as we said, to three decimal places.

Well, as we said when we were looking at what 𝑥 to the power of 𝑎 over 𝑏 was, we can have it written in two ways. To solve this problem, we used the second way. However, just to double check, we could use the first way of rewriting this. And that would be that the square root of 𝑥 all cubed is equal to five over three. So, using this method, what we would do first is take the cube root of both sides of the equation. So, that leaves us with the square root of 𝑥 is equal to the cube root of five over three.

So then, what we do is we’d square both sides of the equation to get our final answer. So, what we get is 𝑥 is equal to, and then we’ve got the cube root of five over three all squared. Now, we can pop this into our calculator. And this would give us our answer. Or using the rule that we looked at earlier with our exponents, we could see that it’ll be 𝑥 is equal to five over three to the power of two over three. And we could type this into our calculator. And either way, we get the correct answer, which is 𝑥 is equal to 1.406. And that’s to three decimal places.