Question Video: Simplifying and Solving Equations Involving 𝑛th Roots Mathematics

Video Transcript

Find the value or values of 𝑥 given that 𝑥 plus nine over five squared equals the square root of 144 times three squared.

Let’s begin this question by seeing if we can evaluate the right-hand side of this equation. We could, of course, work out that three squared is nine, multiply nine by 144, and find the square root. But there is a more efficient way to work this out, particularly if this is a question where we’re not allowed to use a calculator. Let’s use the exponent rule, which tells us that the 𝑛th root of 𝑎𝑏 is equal to the 𝑛th root of 𝑎 times the 𝑛th root of 𝑏 where the 𝑛th of 𝑎 and the 𝑛th root of 𝑏 are real numbers.

And so, the right-hand side of our equation will be equal to the square root of 144 times the square root of three squared. We know that the square root of 144 is 12 and the square root of three squared is three. And multiplying 12 by three gives us 36. The complete equation is therefore 𝑥 plus nine over five squared equals 36.

Next, we can perform the inverse operation of squaring, which is taking the square root. We will have 𝑥 plus nine over five equals plus or minus the square root of 36. Notice how we have included the plus or minus sign. This is because when we’re taking an even root, and we can include the small two in the square root, of a positive value, such as this 36, then we know that there will be two solutions. In this case, we will have the positive and the negative of the square root of 36.

So, the right-hand side of this equation will simplify to plus or minus six. This means that when we continue solving the problem, we have two possibilities: either 𝑥 plus nine over five equals positive six or 𝑥 plus nine over five equals negative six. In each of these equations, the next step in solving them would be to multiply both sides of each equation by five. We would have 𝑥 plus nine equals six times five, which is 30. Or in the other equation, we would have 𝑥 plus nine is equal to negative six times five. That’s negative 30. Subtracting nine from both sides of both equations, we have 𝑥 is equal to 21 and 𝑥 is equal to negative 39.

And so, we can give the answer that there are two values of 𝑥 which satisfy the given equation: 𝑥 is equal to 21 and 𝑥 is equal to negative 39. As a check of our answer, we can substitute these values into the given equation. That means that when 𝑥 is equal to 21, we need to check if 21 plus nine over five squared is equal to the square root of 144 times three squared. On the left-hand side, we would have 30 over five squared is equal to 36. We know that 30 over five is equal to six, so we have six squared is equal to 36. And since both sides of this equation are equal to 36, then we know that 𝑥 equals 21 must be a solution.

In the same way, we can check if 𝑥 is equal to negative 39 is a correct solution by checking if negative 39 plus nine over five squared is equal to the square root of 144 times three squared. This time on the left-hand side, we have negative 30 over five squared. We know that if we square a negative value, we get a positive value. Once again, both sides of this equation are equal to 36. And so, we have verified that 𝑥 equals negative 39 is the second valid solution.