Friday, 2 October 2015
NON PERFET SQUARE ROOT EXAMPLES
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READ MORE:
http://2012books.lardbucket.org/books/beginning-algebra/s04-06-exponents-and-square-roots.html
http://burningmath.blogspot.co.uk/2013/12/finding-square-roots-of-numbers-that.html
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1.6 Exponents and Square Roots
LEARNING OBJECTIVES
- Interpret exponential notation with positive integer exponents.
- Calculate the nth power of a real number.
- Calculate the exact and approximate value of the square root of a real number.
Exponential Notation and Positive Integer Exponents
If a number is repeated as a factor numerous times, then we can write the product in a more compact form using exponential notation. For example,
The base is the factor, and the positive integer exponent indicates the number of times the base is repeated as a factor. In the above example, the base is 5 and the exponent is 4. In general, if a is the base that is repeated as a factor n times, then
When the exponent is 2, we call the result a square. For example,
The number 3 is the base and the integer 2 is the exponent. The notation 32 can be read two ways: “three squared” or “3 raised to the second power.” The base can be any real number.
It is important to study the difference between the ways the last two examples are calculated. In the example (−7)2 , the base is −7 as indicated by the parentheses. In the example −52 , the base is 5, not −5, so only the 5 is squared and the result remains negative. To illustrate this, write
This subtle distinction is very important because it determines the sign of the result.
The textual notation for exponents is usually denoted using the caret (^) symbol as follows:
The square of an integer is called a perfect square. The ability to recognize perfect squares is useful in our study of algebra. The squares of the integers from 1 to 15 should be memorized. A partial list of perfect squares follows:
Try this! Simplify (−12)2 .
Answer: 144
Video Solution
(click to see video)
When the exponent is 3 we call the result a cube. For example,
The notation 33 can be read two ways: “three cubed” or “3 raised to the third power.” As before, the base can be any real number.
Note that the result of cubing a negative number is negative. The cube of an integer is called a perfect cube. The ability to recognize perfect cubes is useful in our study of algebra. The cubes of the integers from 1 to 10 should be memorized. A partial list of perfect cubes follows:
Try this! Simplify (−2)3 .
Answer: −8
Video Solution
(click to see video)
If the exponent is greater than 3, then the notation an is read “a raised to the nth power.”
Notice that the result of a negative base with an even exponent is positive. The result of a negative base with an odd exponent is negative. These facts are often confused when negative numbers are involved. Study the following four examples carefully:
| The base is (−2) | The base is 2 |
|---|---|
The parentheses indicate that the negative number is to be used as the base.
Example 1: Calculate:
a. (−13)3
b. (−13)4
Solution: The base is −13 for both problems.
a. Use the base as a factor three times.
b. Use the base as a factor four times.
Answers: a. −127 ; b. 181
Try this! Simplify: −104 and (−10)4 .
Answers: −10,000 and 10,000
Video Solution
(click to see video)Square Root of a Real Number
Think of finding the square root of a number as the inverse of squaring a number. In other words, to determine the square root of 25 the question is, “What number squared equals 25?” Actually, there are two answers to this question, 5 and −5.
When asked for the square root of a number, we implicitly mean the principal (nonnegative) square root. Therefore we have,
As an example, 25−−√=5 , which is read “square root of 25 equals 5.” The symbol √ is called theradical sign and 25 is called the radicand. The alternative textual notation for square roots follows:
It is also worthwhile to note that
This is the case because 12=1 and 02=0 .
Example 2: Simplify: 10,000−−−−−√ .
Solution: 10,000 is a perfect square because 100⋅100=10,000 .
Answer: 100
Example 3: Simplify: 19√ .
Solution: Here we notice that 19 is a square because 13⋅13=19 .
Answer: 13
Given a and b as positive real numbers, use the following property to simplify square roots whose radicands are not squares:
The idea is to identify the largest square factor of the radicand and then apply the property shown above. As an example, to simplify 8√ notice that 8 is not a perfect square. However, 8=4⋅2 and thus has a perfect square factor other than 1. Apply the property as follows:
Here 22√ is a simplified irrational number. You are often asked to find an approximate answer rounded off to a certain decimal place. In that case, use a calculator to find the decimal approximation using either the original problem or the simplified equivalent.
On a calculator, try 2.83^2. What do you expect? Why is the answer not what you would expect?
It is important to mention that the radicand must be positive. For example, −9−−−√ is undefined since there is no real number that when squared is negative. Try taking the square root of a negative number on your calculator. What does it say? Note: taking the square root of a negative number is defined later in the course.
Example 4: Simplify and give an approximate answer rounded to the nearest hundredth: 75−−√ .
Solution: The radicand 75 can be factored as 25 ⋅ 3 where the factor 25 is a perfect square.
Answer: 75−−√≈8.66
As a check, calculate 75−−√ and 53√ on a calculator and verify that the both results are approximately 8.66.
Example 5: Simplify: 180−−−√ .
Solution:
Since the question did not ask for an approximate answer, we present the exact answer.
Answer: 65√
Example 5: Simplify: −5162−−−√ .
Solution:
Answer: −452√
Try this! Simplify and give an approximate answer rounded to the nearest hundredth: 128−−−√ .
Answer: 82√≈11.31
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