Using Radicals to Solve Equations

The lesson below is still images taken from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. If the PowerPoint is not one of my freebies, you may also head over to my YouTube channel to see the slideshow fully animated and can pause as needed to be sure you grasp each concept before moving forward (each lesson will always be free there!).

In this lesson we go over how to solve equations with exponents by using radicals to “cancel” them out. The slides first remind students how to use inverse operations to solve for variables. Then we discuss how the radical is inverse to raising a variable to a power.

So without further ado, read through the slides below to get a feel for how to solve equations using radicals!

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Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

The Pythagorean Theorem

Just in case the slides aren’t your thing, here is a text outline of the main points of the lessons above!

  • Radicals
  • Solving equations
  • Objectives
  • By the end of this lesson you should feel comfortable:
  • Using radicals to solve equations with exponents
  • Equations
  • Remember equations are expressions separated by an equal sign.
  • We practiced solving for an unknown quantity, a variable, in equations such as these:
  • We simply use an inverse operation to isolate the variable, or get it by itself.
  • However, what happens if we see an exponent in our equation?
  • Equations with Exponents
  • It is possible to see a variable raised to a power, as we have seen.
  • This means it must also be possible to see such a thing in an equation!
  • This is read as “x squared equals thirty-six.”
  • But it is easier to find a solution if we think along the lines of “Something squared equals 36.”
  • What times itself is equal to 36?
  • Of course! 6 x 6 = 36
  • Equations with Exponents
  • Alright, hopefully not too bad.
  • But why do we need radicals if problems like these can be solved like this?
  • Good question!
  • It’s sort of like division and multiplication facts.
  • No need to remember “division facts” when you can ask “What times 3 equals 15?” to solve 15 divided by 3.
  • Yet people still remember 15 divided by 3 is 5, and we do still use division for things rather than multiplication to make life easier.
  • Using square roots
  • Well, what about this?
  • Do you know off the top of your head quickly what times itself equals 625?
  • Maybe you remember perfect squares this high, but I certainly didn’t until I had been teaching math for a while.
  • I hear you, though. We could guess and check trying to multiply numbers together until we figured out 25×25=625.
  • How about a number that isn’t a perfect square then?
  • This answer will be irrational, so how can we find x?
  • Using Square roots
  • First, let’s try a simpler problem to explain the method.
  • Perfect, back where we started.
  • We said this could be solved by rephrasing the question as “What times itself equals 36?”
  • But isn’t that exactly how we evaluated square roots?
  • So how did we transform our equation from the top to the bottom one?
  • Inverses!
  • Using Square Roots
  • The inverse of addition is subtraction, and vice versa.
  • The inverse of multiplication is division, and vice versa.
  • So then, as you may have guessed, the inverse of squaring a number is finding its square root!
  • x times x is x squared, just like any number, so to “undo” x squared and make it “x” we just have to take the square root:
  • But if we do something to one side of the equation, we have to do it to the other side so things stay balanced:
  • And of course, the square root of 36 is 6, so our answer is:
  • Using Square Roots
  • Let’s see a couple more just to be sure we understand how to solve these.
  • Beginning with the top equation, how do we “cancel” the square?
  • And keeping both sides balanced:
  • So our new equation looks like:
  • Then evaluate the square root:
  • Perfect! Now see if you can follow along with the bottom equation.
  • Other Exponents
  • We know that any number can be an exponent, though.
  • So how would we go about solving an equation like this?
  • Or this?
  • Well, we use square roots for squares because the index matches the exponent.
  • In the first example, our exponent is 3, so which root has an index of 3?
  • Now it’s the same process as with square roots!
  • Other exponents
  • Our new equation:
  • And what times itself times itself equals 64?
  • Tada!
  • Now what about the next one?
  • Our index needs to match our exponent, so we will need to take a 10th root to isolate x.
  • So our answer with a little work would be:
  • Now you’re ready to solve any one-step equation with an exponent!
  • conclusion
  • Equations can have variables risen to exponents.
  • To solve for the variable, we use roots to “undo” the exponents.
  • The square root is the inverse operation of squaring a number (raising it to the second power).
  • The inverse of any nth power is its corresponding nth root.
  • It is possible to have an irrational answer to an equation with an exponent when taking a root of a number that isn’t a perfect square, cube, or higher corresponding root.