E - Square and Cube Roots & Rational Approximations Lesson

Square and Cube Roots & Rational Approximations

From the previous lesson on exponential expressions, you'll remember that when you raise a number to the second power, we call that squared. If something is raised to the third power, we say that number is being cubed.

In this lesson, we'll be focusing on numbers that are being squared and cubed and we'll pay particular attention to their inverses. The inverse is often thought of as "opposite operation." An inverse operation is the operation that "undoes" another operation. For example, addition and subtraction are inverse operations. Likewise, multiplication and division are inverse operations. The inverse of squaring a number is taking its square root. Similarly, the inverse of cubing a number is taking its cube root. This is true for all nth roots.

Let's look at the notation for nth roots. An nth root's notation has 3 parts: the radical, which is just the root symbol, the index, which tells you which root you are taking, and the radicand, which is the information under the radical.

Let's look at this table that will help you see a list of some of the most commonly used square and cube roots. These numbers that have a square root and a cube root are called perfect squares or perfect cubes.

For example, 4 is a perfect square because (2)(2)= 4. The square root of 4 is 2.

You probably noticed from this table that not every number is listed here. That is because not all numbers are perfect squares or perfect cubes.

Every positive number has two square roots, one positive and one negative.

One square root of 16 is 4, since 4•4= 16.

The other square root of 16 is -4, since (-4) • (-4) is also 16.

You can write the square roots of 16 as ±4, meaning "plus or minus" 4.

Find all square root solutions for the equations.

Think to yourself, what are all possible values of x. In other words, for example, what squared would give me 49?

  x²= 49

Here, since we are solving an equation, our answer would be ±7, because (7)(7)=49 and (-7)(-7)=49.

You try the next two. Work through the equation, then check the solution.

• Question: $x^2=25$
  
  • Solution: +$5$
• Question: $x^2=\frac{4}{9}$
  • Solution: $\pm\frac{2}{3}$ 

Let's take a look at some where we are simply given a number, not an equation in which to give the square root.

$-\sqrt[]{49}$

Since the negative sign is on the outside, we treat that as a factor of -1. We will multiply that factor of -1 by the square root of 49. In other words, (-1)(7) = -7.

You try.... solve the following equations.

Question: $-\sqrt[]{\frac{1}{16}}$

• Solution: $-\frac{1}{4}$

Question:  $\sqrt[]{.09}$

• Solution:  .3

Here are some problems for you to practice!

HERE is a link to learn about Non-Perfect Squares. Links to an external site.

Homework

Now that you have spent some time learning about square roots, you are ready to complete your Homework. Click here to download the Scientific Notation Homework. Links to an external site.

Once you have completed the Square and Cube Roots & Rational Approximations Homework, check your answers and make sure you ask your teacher if you have any questions. Links to an external site. When you feel confident in your work, you'll need to take the Square and Cube Roots & Rational Approximations Homework Quiz. Make sure you have your Homework completed before you log in to take it and remember this is a timed quiz.

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