# Equations with Indices – Corbettmaths

Hi welcome to score math video on equations in this video, we’re gonna, look at how to solve equations which involve indices or roots! So let’s have a look at our first question. Our first question says: solve x to the power of 1/2 equals 8 now to solve an equation.

We want to do the inverse operations to the both sides until we’re left with x equals a certain number. So here we’ve got x to the power of 1/2. Now the power of 1/2 means two square roots, so we would want to do the opposite of the inverse operation so, instead of square root, n we’re gonna square both sides of this equation, so squaring the left-hand side. Well, just leave us with x and squaring the right-hand side will give us what 8 squared is 64. So our answer would be x, equals 64. At the check our answer, we could just substitute the 64 in so 64 to the power of 1/2! What’s the square root of 64, which is it so that’s right. So if you’ve got an equation where it’s x to the power of 1/2 to solve it, you would just square both sides of the equation if it was x to the power of 1/3. Well, that means the cube root of x, but then we would cube both sides of the equation if it, if it was x to the power of 1/4 i mean so for fruits, so we would do to the power 4 to both sides of the equation. So if you’ve got an equation, where you’ve got x to the power of 1 over something, you would just take that power. The denominator in you did that power to both sides of the equation and you a solve your equation?

Ok, let’s have a look at a question that, where it’s not a 1 on the numerator, so our next question solve x to the power 3/4 equals 8 now x to the part of 3/4, that’s the same as saying the fourth root of x and then cubing it. So whenever we want to solve this equation, we monitor the inverse.

So, instead of doing the cubing, we’re gonna cube root! Both sides of the equation and then i said of the fourth root: we’re gonna do the power 4 to both sides of the equation! So, first of all, let’s do the cube root the opposite of this cubing! So taking the cube root, look at the numerator, it’s a phrase. We’re gonna, take the cube root of both sides of the equation and so taking the cube root of the left-hand side, but i would just be x to the power of 1/4, we’re doing it to get rid of the 3 on the numerator and the cube root of the left-hand side. Well, the cube root of 8 is equal to 2. Now it’s just with this part of the equation is just like our last question? If we wanted to solve this, we wanna do the inverse operation of the four fruits, so that would be to the power of 4. So that would leave us with x? + 2 to the power 4 to to the par 4 is equal to 2 times 2 times 2 times, 2 well 2 times, 2 is 4 times. 2 is 8 times, 2 is 16, so answer would be x, equals 16 and again we can check our answer! 16 by the 4th root of 16 is equal to 2 and cubed is equal to it and that’s it alright.

So our next question it says: solve x to the power of negative 2 equals 9. So this questions got a negative power, so we’re going to have to be careful whenever we’re solving this so x to the power negative 2! What’s the same as 1 over x squared when you’ve got a negative power, you can write it as 1 over or the reciprocal. Now we’ll come back to that later. On the question to show you how you can sort of to this question using a slightly different approach, so we’ve got 1 over x, squared equals 9! Now we’re going to this we’re going to multiply both sides to the equation by x squared so they give us one equals and 9 x squared and then we’re gonna divide both sides of the equation by 9 to get 1/9 equals x. Squared now we’re going to be careful here, because what we’re going to do is we’re going to square both sides, but we’ve got to be careful because we’ve got x, squared equals 1/9! Now a positive squared is a positive, but also negative! Squared is equal to a positive. So whenever we square both sides, we need to remember that we could have the positive or the negative solution. Whenever you got x, squared equals a number a positive number and your square root in it. You could have the positive or negative roots, so we have got the square root of 1/9, and so the square root of the 9 for our square root of 1 is 1 and square root of 9 is equal to 3.

So that means that x will be equal to positive or negative 1/3 i’m, just gonna write out and fill the x equals, 1/3 or x equals negative 1/3, and we can check our answers so we had 1/3 and we done to the part of negative 2. That would be the reciprocal of that?

Well, the reciprocal of 1/3 is equal to 3 and squared is 9 and that’s our solution there or if we had negative preferred well, the reciprocal of that would be negative? 3 and squared would be 9? So there’s our two solutions. Now there was a slightly different approach.

We could have taken on that question rather than starting off by writing 1 over x squared! So what we could have done was we could have taken the reciprocal of both sides, speaking with to get rid of the negative son, because the negative sign means 1 over. If we get rid of that, we can just say well x, squared that’s taking the reciprocal of it is equal to 1, and that could mean that we could just go straight to this part here and then we could square root both sides, remembering it’s the positive or negative solutions. Okay, our next question: our next question is to solve the cube root of 9x. Minus 1 equals 4!

So again we want to do the inverse operations to both sides? This is the cube root, so we’re going to cube both sides of this equation, so keeping the left hand side. You just leave us with 9x minus 1 and cubing the right hand.

Side would be 64! Now we need to find what 9x is so we’re gonna, add 1 to both sides, so 9x equals 65 and then dividing by 9 will give us x, equals 65 over 9 and so x equals 65 over 9 and that doesn’t simplify. So that’s our solution and that’s it! Okay? Next question: our next question is a little different because we have x to the power three-halfs, but then we’ve got a mixed number here of 4 and 17 over 27. What we’re gonna do is we’re gonna write this as a top-heavy fraction to begin with, and that might help us, so we’ve got x to the power of 3 over 2 all right. This is a top-heavy fraction about 4 times, 27 well 4 times, 27 is equal to 108, plus 17 is equal to 125 and then that’s gonna be over. The denominator stays the same? So it’s 27 now to solve this.

Remember we’re going to take whatever it’s on the numerator! We’re gonna take that rich of both sides? So we’ve got a cubed here, so we need to do the inverse, which is the cube root of both sides, and that’s quite nice, because both of the numbers are cube numbers. So it’s gonna, give us x to the 1/2. Getting rid of the 3 will equal and taking the cube root of both of these numbers would give us well.

The cube root of 125 is 5 and the cube root of 27 is equal to 3 and then finally, we’ve got x to the power of 1/2, which is the square root. So we need to square both sides of the equation, so squaring both sides will give us x equals 25 over 9, and that’s it right. So our last two question is a little bit different than the ones we’ve done so far because they involve the laws of indices, so you’ve got our first question says: solve 3 to the power of 4x equals 27 to the power of 5 minus x!

Now, in this question, we’ve got our free and our 27.

Now these numbers aren’t chosen at random and they’re chosen because 27 is equal to 3 cubed. So what we’re going to do is we’re going to rewrite the 27 as free cubed, so we’ve got 3 ^ 4 x equals 3, cubed ^, 5 minus x!

Now what we’re going to do this we’re going to use the laws of indices here, so we’ve got a power of a power so whenever you’ve got x to the power of a all to the power of b, that would be the same as x to the a b? So you multiply the end, the b! So if we multiply the 3 this cubed by the 5 – x, then we can write it as 3 ^, 15, -, 3 x, so will give us, on the left hand, side 3 to the power of 4x equals and multiplying the 3 here, the cubed by the 5 and the minus x? Where goes free to the power of 15 – 3 x, now we’ve got 3 to the power of something equals 3 to the power of something submitted! The – something’s must be the same, so in other words, that must give us 4x equals 15, – 3 x never solve this. We can just add 3x to both sides of the equation, so that’ll be 7x, equals 15 and dividing by 7 gives us x equals 15 over 7! That’s it! so if we’ve got a question where you’ve got the 3 and the 27, you can write, the 27 are 3 cubes and then you use the laws of indices of silver!

Let’s have a look at a similar question, i so similar, but a little bit different. We’ve got our question where it says solve it: to the power of 4 plus x, over 4 to the power of 5.

– x equals no point 5. Now? The first thing i’m going to do is i’m going to rewrite the null point!

5 is 1/2, so we’re gonna, write x or it’s the power of 4 plus x over 4 to the power 5 – x equals 1/2. Now all of these, the it the 4 and 1/2 are all numbers which can be written as powers of 2!

It is the same as 2 cubed 4 is obviously 2 squared and 1/2.

Well, that’s the same as 2 to the minus 1. So i’m going to rewrite this as 2 cubed to the power of 4 plus x, over 2 squared to the power of 5 – x, equals two to the negative one. So then, what i’m not going to do is well! Are you gonna make a bit of space and then we’re going to go to do is i’m going to use the laws of indices here, because these are powers over power, so i’m going to multiply these two powers together, so it’ll be 2 to the power off 3 times! 4 is 12 and 3 times.

X is plus 3 x over and then x in the powers together here will give us 2 to the power of or 2 times, 5 is equal to 10 and 2 times minus x. We minus 2 x that equals 2 to the negative 1! Now here we’ve got 2 to the power of something divided by 2 to the power of something again using the laws of indices? If you had x to the power of a divided by x to the power of b, that’s the same as x to the power of a minus b.

You take away the powers so here if we have 2 to the power of 12 plus 3x over 2 to the power of 10 minus 2x. If we take away the powers, we can write it as 2 to the power of something. So let’s take away the powers and see what we get. So if we had 12 plus 3x and we subtract from that 10 minus 2x or that will give us well 12, take away. 10 is 2 +, 3, x, -, minus 2x. Will that be plus 5x? somebody said the left hand?

Side would have become 2 to the power of 2 plus 5 x and that equals 2 to the negative 1? Now, if you got 2 to the power of something equals 2 to the power of something well, that makes it 2 for something’s must be equal to each other. That means a 2 + 5 x must be the same as minus 1. So let’s write that down.

2 plus 5x is equal to negative 1, so a desk just ticked away from both sides of the equation so to be 5 x, equals a negative 3 and dividing by 5 would give us x equals negative 3/5. That’s it. ?