Algebra II : Irrational Numbers
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Example Questions
Example Question #11 : Irrational Numbers
Which irrational number is between 9 and 10?
Since all the possible answers are square roots, we can square both the limits of our problem and all the possible answers. This allows us to see which number is correct. After squaring everything, we notice that we need a number between 81 and 100. The only possible answer given is 90. Now we can square root everything back, giving us our final answer of 
Example Question #12 : Irrational Numbers
Rationalize:
In order to rationalize, we will need to multiply the top and bottom by the denominator in order to eliminate the square root in the denominator.
Distribute and simplify the numerator. Multiplying unlike numbers inside square roots will not eliminate the square root!
The answer is:
Example Question #52 : Number Theory
Which of the following is equal to
Simplify the radicals
We notice that we have a complex number in the denominator. To get rid of this we multiply the numerator and denominator by the complex conjugate of the denominator.
Distribute across the numerator and multiply the binomials in the denominator. You may use the FOIL method.
We know that 
Combine like terms
Reduce and put in standard form
Example Question #51 : Number Theory
Find the differance
simplify the radicals
Distribute the negative
Combine like terms
Example Question #12 : Irrational Numbers
Which set does NOT contain an irrational number?
Irrational numbers are nonrepeating decimals-- they cannot be written as fractions.
Example Question #55 : Number Theory
Which of the following is equivalent to
Factor the number -96
-1 2 2 2 2 2 3
The -1 come out the radical as an i. We look for pairs in the factors.
-1 ( 2 2 )( 2 2 ) 2 3
We see two pairs of 2, both can come out from under the radical .
Multiply
Example Question #13 : Irrational Numbers
What is the complex conjugate of 
To find a complex conjugate we change the sign of the imaginary part of the number.
To be in standard form the real number should come before the imaginary part of the number
Example Question #57 : Number Theory
Solve the following equation for x. Express your answer with complex numbers.
Our first goal is to isolate the X. So we subtract the 3 on both sides.
Now we divide by 2 on both sides
We square root both sides
Simplify the radical
Example Question #15 : Irrational Numbers
What is the complex conjugate of
The square of 324 is 18. The negative under the radical means that
our problem is now
Example Question #16 : Irrational Numbers
Solve the following equation. Express your answer with complex numbers.
None of the other answers are correct.
We need to isolate the x. First we subtract 21 on both sides.
We now take the square root of both sides
The square root of a negative has an i
Now just add the 7 to both sides
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