Radicals

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 evaluate roots using radicals. The slides first introduce what a radical is as well as the index and radicand. After covering the vocabulary we discuss how to solve square roots using perfect squares and then transition into roots with large indices. Finally, students will wrap up the lesson by seeing how to estimate square roots by placing them between perfect squares.

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

Rad 1Rad 2Rad 3Rad 4Rad 5Rad 6Rad 7Rad 8Rad 9Rad 10Rad 11Rad 12

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!

Using Radicals to Solve Equations

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

  • Radicals
  • Evaluating Roots
  • Objectives
  • By the end of this lesson you should feel comfortable:
  • Identifying radicals and roots and evaluating them
  • Understanding radicals as rational or irrational numbers
  • Radicals
  • To begin, what is a radical?
  • Well, it’s this thing:
  • You may have heard of this as a root or square root sign before.
  • However, it’s official name is a radical.
  • Radicals can appear in a number of ways:
  • Roots
  • Alright, so now that we know what this new symbol is called, what does it do?
  • Remember exponents?
  • 5 to the second power, or 5 squared, means 5 x 5 = 25.
  • But like good mathematicians, we should ask ourselves, can I do this in reverse?
  • Yes!
  • Ok…So exactly what is happening here?
  • Roots
  • Per usual, let’s look at this piece by piece.
  • The radical symbol is asking us to find a root, or what multiplied by itself some number of times equals that number.
  • The index is the number on the top left of the radical symbol.
  • This tells us what root we are taking, or how many times a number should be multiplied by itself to equal the number beneath, called the radicand.
  • In this case we are finding the 2nd root, or square root, of 25.
  • As a side note, we typically do not write the 2 and instead just understand that a radical with no index is a square root (lazy mathematicians!):
  • Roots
  • Ok that was a lot of fancy math talk and vocabulary.
  • Let’s try figuring out how we go about solving these roots.
  • Because we have the square root of 25 here, another way of thinking about this problem is “What times itself is 25?”
  • We should hopefully remember 5 x 5 = 25, so 5 is our answer!
  • How about another?
  • So what times itself is 64?
  • Perfect, 8 x 8 = 64!
  • nth Roots
  • So now that we’ve experienced square roots, let’s raise the stakes a bit.
  • What’s different?
  • This is no longer a square root. Instead, because of the 3, we have a 3rd root, or cube root.
  • What does this change in our calculation?
  • Remember this is called the index and tells us how many times a number should be multiplied by itself to get our radicand.
  • In this case, our index is 3, so what times itself times itself is 64?
  • Perfect! 4 x 4 x 4 = 64.
  • Nth Roots
  • So higher roots can follow this pattern.
  • The 4th root of 16:
  • What times itself times itself times itself is 16?
  • The 5th root of 3,125:
  • What times itself times itself times itself times itself is 3,125?
  • And so on and so forth as high as you want or need your index to be!
  • I’ll admit, that fifth root forced me to use a calculator since I didn’t know off the top of my head what to the fifth power was equal to 3,125.
  • Which actually brings me to my next point…
  • Estimating Roots
  • All of the examples we have worked through so far have had nice, whole number roots.
  • Alas, we know this cannot be the case for all roots.
  • What is different about these radicands and those we saw before?
  • The square roots we took before had perfect squares under the radical.
  • Perfect squares are always something squared.
  • So what about this guy?
  • Estimating Roots
  • Well, if you plugged this into a calculator you would get:
  • Hmm, an irrational answer.
  • Remember that irrational numbers are numbers that are not rational.
  • So they have decimals that never end, or terminate, and also never repeat anywhere even though they go on forever.
  • I’ll be the first to tell you, slapping that fella into a calculator is your best bet when you need decimals.
  • But most of the time we can get by just knowing about what an irrational root is.
  • Estimating Roots
  • So how can we look at this monster and get a feel for what decimal it is without a calculator?
  • Since the square root of perfect squares is much easier to evaluate, we will use those.
  • 20 is between two perfect squares, but which two?
  • Well, let’s go back to our old friend the number line.
  • We now know that if we square each number, then it gives us a perfect square, so let’s rewrite those on the top:
  • Ah, and 20 is between 16 and 25 just a little closer to 16:
  • Estimating Roots
  • Maybe a couple more examples will help clear it up.
  • First, the square root of 7.
  • 7 is between which two perfect squares?
  • 4 and 9! And what are the square roots of these perfect squares?
  • Since 7 is a little closer to 9, our answer should be a little closer to 3. Is it?
  • Perfect! See if you can follow along yourself for the next one:
  • conclusion
  • A radical is a symbol for the inverse of evaluating exponents.
  • A radical has an index on the top left of the symbol and a radicand inside.
  • The answer to a radical expression is called a root.
  • Radicals ask the question “What multiplied by itself index times is equal to the radicand?”
  • A radical with no index is understood to be a square root. (index of 2)
  • An index of 3 is called a cube root.
  • Larger roots are just referred to as nth roots. (4th root, 5th root, 6th root…..)
  • Perfect squares are numbers that have whole number square roots.
  • We can estimate irrational square roots by placing them between perfect squares.
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