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Introduction to Square Roots

Consider √x. This is read as, "the square root of x." In this particular term, x is called the base of the square root.

Basic square roots have no number written on the root, and are assumed to be the second root of the base. So, when solving for the square root of x, we want to know what other number multiplied by itself two times will result in x.

For instance:

√9 = 3, because 3 x 3 = 9.

√25 = 5, because 5 x 5 = 25.

√16 = 4, because 4 x 4 = 16.

A common mistake when calculating square roots is to divide the base by two. For instance, in the last example, a student may say that √16 = 8, because 16/2 = 8. Take care! Finding the square root is not dividing by 2, but rather what number multiplied by itself will result in our base.

All of the examples so far have used perfect squares, or numbers for which there is a perfect, integer square root. This is not always the case. We can easily estimate the value of such a problem.

For instance:

√20

This base is not a perfect square. If we input this term into the calculator, we will get an irrational number that would need to be rounded.

However, we do not need a calculator to get a pretty good guess for the value of this expression. Consider:

√16 = 4

√25 = 5

16 < 20 < 25

Our answer must be between 4 and 5, because our base is between the perfect squares 16 and 25.

PRACTICE PROBLEMS

1. Consider the term √36.

a. What is the base?

b. What is the answer?


2. Consider the term √43.

a. What is the base?

b. Estimate the answer.


3. Andrew worked a problem involving square roots. His work is shown below:

√100 + √64 = 50 + 32 = 82

Explain what Andrew did wrong.

ANSWERS TO PRACTICE PROBLEMS

1.a. The base is 36. 1.b. √36 = 6, because 6 x 6 = 36.

2.a. The base is 43.

2.b. Since 43 is not a perfect square, estimate the answer based on the perfect squares directly before and after 43. 36 is the perfect square before 43, and √36 = 6. 49 is the perfect square after 43, and √49 = 7. So, √43 must be between 6 and 7.

3. Andrew is finding the number that yields the base when multiplied by two rather than by itself. We cannot divide by two when finding a square root. Instead:

√100 = 10, because 10 x 10 = 100

√64 = 8, because 8 x 8 = 64

So √100 + √64 = 10 + 8 = 18


Related Links:
Math
algebra
Dividing with Square Roots
Introduction to Cube Roots
Simplifying Square Roots
Multiplying Square Roots with Exponents


Adding Square Roots Worksheets
Simplifying Square Roots Worksheets




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