Radicals - Schenectady Math Portal



Radical Expressions

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Name: _________________________ Period:__________

College Bound Math Teacher:______________

Reference Sheet

|Axis of symmetry: |[pic] |

|Quadratic Formula: |[pic] |

|Determinate of a 2×2 matrix |[pic] |

|Inverse of a 2×2 matrix |[pic]( [pic] |

|I = interest B = Balance P = principle |

|r = rate (as a decimal) n = number of compounding periods t = time in years |

|Exponential growth & decay |B = P(1 + r)t B = P(1 – r)t |

|Simple Interest |Compound Interest |Continuous Interest |Future Value – periodic |Present Value – one time|Present Value - periodic |

|[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |

|Key Words: |Key Words: |Key Words: |Key Words: |Key Words: |Key Words: |

|Simple interest |Compounded |Compounded |Total balance |Deposit now |Deposit now |

| |Annually, semiannually, |continuously |Each/every |Starting principal |Starting principal |

| |quarterly, etc. | | |goal |Each/every |

| | | | | |goal |

|Trigonometric Ratios |Sin A = [pic] Cos A = [pic] Tan A = [pic] |

|Coordinate Geometry |[pic] |

|Slope – Intercept Formula |y = mx + b |

|Law of Sines |[pic] |

|Law of Cosines |(side) [pic] |

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| |(angle) [pic] |

|Area of a Triangle |[pic] |

Lesson 1 Simplifying Radicals

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|You use a radical sign to indicate a root. The number under the radical sign is called the | |

|_________________________. The _________________ gives you the degree of the root. | |

Examples:

a.) What is the radicand?

b.) What is the index?

c.) What is the answer?

|1.) [pic] |2.) [pic] |3.) [pic] |4.) [pic] |

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|a.) ________________ |a.) ________________ |a.) ________________ |a.) ________________ |

|b.) ________________ |b.) ________________ |b.) ________________ |b.) ________________ |

|c.) ________________ |c.) ________________ |c.) ________________ |c.) ________________ |

|5.) [pic] |6.) [pic] |7.) [pic] |8.) [pic] |

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|a.) ________________ |a.) ________________ |a.) ________________ |a.) ________________ |

|b.) ________________ |b.) ________________ |b.) ________________ |b.) ________________ |

|c.) ________________ |c.) ________________ |c.) ________________ |c.) ________________ |

What is i?

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Sometimes, the answer will NOT be a perfect square, but that doesn’t mean we can’t simplify it in lowest radical terms.

|Example 1: [pic] |Example 2: [pic] |

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Try a few on your own – pull out the LARGEST PERFECT SQUARE as soon as possible, otherwise you will end up doing more work! Feel free to list the perfect squares down the side of the page!

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We can also simplify radicals that have variables in the radicand. Just look at the exponents!

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|8.) [pic] |9.) [pic] |10.) [pic] |

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What pattern are you noticing? ______________________________________________________

Now you try a few yourself!

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|13.) [pic] |14.) [pic] |

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Practice – try the following questions to practice simplifying radicals.

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Lesson 2: Pythagorean Theorem – Simplifying radicals

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Lesson 3: Adding and Subtracting Radical Expressions

Think back to polynomials – how to I add/subtract the following?

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Just like with polynomials, in order to add or subtract, we need _______________________!!

Are the following “like radicals”?

|1.) [pic] and [pic] |2.) [pic] and [pic] |3.) [pic] and [pic] |

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|4.) [pic] and [pic] |5.) [pic] and [pic] |6.) [pic] and [pic] |

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Add or subtract the following radicals. Simplify first if necessary.

|7.) [pic] |8.) [pic] |9.) [pic] |

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|10.) [pic] |11.) [pic] |12.) [pic] |

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Practice – practice simplifying, adding and subtracting the following radicals.

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Lesson 4: Multiplying Radical Expressions

We multiply radical expressions very similarly as we multiply polynomials. Comparing them will be helpful.

|Polynomials |

|[pic] |[pic] |[pic] |[pic] |

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|Radicals |

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Classwork/Practice:

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Lesson 5 Dividing Radical Expressions

We do not typically leave radicals in the denominator of a fraction, so when we “divide” radical expressions, we multiply to _______________________ the radical in the bottom.

|Example: |

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Sometimes, it may be easier to simplify first, and then get rid of the radical. Either path will get you to the same answer, so pick which way you prefer.

|Example: |

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**Multiply by the radical

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Lesson 6 Solving Radical Equations

A radical equation is an equation that has a variable in a radicand or a variable with a rational exponent.

Solving a square root equation may require that you _______________ each side of the equation. This can introduce extraneous solutions (extra solutions that don’t actually work).

|Example #1: |Check: |

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|Example #2: |Check: |

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|Example #3: |Check: |

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Practice:

|1.) [pic] |2.) [pic] |3.) [pic] |

|4.) [pic] |5.) [pic] |6.) [pic] |

|7.) [pic] |8.) [pic] |9.) [pic] |

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Lesson 7 Imaginary Numbers

A2.N.6 Write square roots of negative numbers in terms of i

In 1 – 24, express each number in terms of i, and simplify.

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic] 16. [pic]

17. [pic] 18. [pic] 19. [pic] 20. [pic]

21. [pic] 22. [pic] 23. [pic] 24. [pic]

A2.N.7 Simplify powers of i

In 1 – 15, write each given power of i in simplest terms as 1, i, –1, or –i. Show how you arrived at your answer.

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|4. [pic] |5. [pic] |6. [pic] |

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A2.N.9 Perform arithmetic operations on complex numbers and write the answer in the form a + bi

In 1 – 13, write each number in terms of i, perform the indicated operation, and write the answer in simplest form.

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|13. Express in terms of i the sum [pic] |

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Lesson 8 Quadratic Formula – Leave in simplest radical form

Solve each quadratic & leave

the answer in simplest

radical form.

x2 - 8x + 16 = 0 x2 + 10x = 24

4k2 - 13 = 12 6x2 - x = 12

5n2 - 24 = 2n x2 + 3x - 6 = 0

x2 + 10x + 8 = 0 8x2 + 8x - 1 = 0

3x2 +4x - 12 = 0 8x2 + 72 = 0

x2 - 4x - 9 = 0 x2 - 4x + 9 = 0

x2 - 3x + 16 = 0 x2 -8x + 28 = 0

x2 - 4x - 5 = 0 x2 - 4x + 5 = 0

2x2 + 7x + 6 = 0 2x2 + 7x - 6 = 0

3x2 + 4 = 6x Simplify i357409

x2 - 3 = 2(4x + 6) (x - 3)2 = 2x - 5

Find three consecutive positive odd integers such that the product of the first and third is 1 less than twice the second.

Find three positive consecutive integers such that the product of the first and second is 2 more than 9 times the third.

Find two negative consecutive odd integers such that their product is 63.

James is four years younger than Austin. If three times James’ age is increased by the square of Austin’s age, the result is 28. Find the ages of James and Austin.

If the length of one side of a square is tripled and the length of an adjacent side is increased by 10, the resulting rectangle has an area that is 6 times the area of the original square. Find the length of a side of the original square.

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