_____(0-10 pts) Describe how to simplify a radical expression. Include a discussion about the properties of radicals. When do we use radicals in real life? Give at least 3 examples of each.

 

 

     To simplify a radical expression, you first have to see if the number under the square root sign is a perfect square. If it is, find the square root of that number. If it isn't, find the greatest perfect square factor of the number. The square root of the other number will probably be a number with decimals, so leave it as it is.

 

 

Product Property of Square Root: When you have a square root with a and b being multiplied together inside (√4x25) it would equal the same as the square root of a and the square root of b separated being multiplied (√4 x √25). This apply to all the positive numbers that are real.

 

 

Quotient Property of Square Root: When a and b are being divide together inside a square root (√36/4) it would as the square root of a was being divided as the square root of b (√36 / √4).

 

 

Radicals can help us in real life too. We can use them to figure out problems that we face in our everyday lives. An example is to find the height of a rectangle with a base of 400 ft^2. We can use our rules to simplify radicals so this problems become easier to solve and so we finally get the answer.

 

Imagine your boat is traveling north from shore. You're father's boat left at the same time going east. After an hour, your father calls you and tells you that he has stopped because of some engine trouble. How far must you travel to meet your father? (You're boat has traveled 20 miles, your father boat has traveled 25 miles)

 

 c=√a^2 + b^2

= √(20)^2+(25)^2

= √400+625

= √1,025

= √41(25)

= 5√41 miles : 32 miles

 

 

 

 

5e√41 miles : 32 miles

A baseball diamond is a square with sides of 90 feet. How far is the throw from first base to third base?Give the answer as a radical expression in simplest form. Then estimate the length to the nearest tenth.

 

c = √a^2 + b^2

= √(90)^2 + (90)^2

= √8100 + 8100

= √16,200

= √100(81)(2)

= √100 √81 √2

= 10(9)√2

= 90√2

estimate of 127.3 ft

 

So the distance is 90√2 or about 127.3 ft.

 

 

Product Property of Square Roots examples:

 

√4000 =   √400√10  = 20√10

 

 

 

√125 = √25√5 = 5√5

 

 

 

 

√180 = √9√20 = 3√20

 

 

 

 

√648 = √4√162 = 2√162

 

 

 

 

Quotient Property of Square Roots examples:

 

 

√6/49  =  √6/√49 = √6/7 

 

 

√512/81 =  √512/√81 =  16√2/9

 

 

√204/25 =  √204/√25 =  √4√51/5 =  2√51/5

 

 

√4x/36x =  √4x/√36x =  2x/6x

 

 

 

 

Simplifying Ordinary Radicals

 

√18= √9 √2 = 3√2

 

√12= √4 √3= 2√3

 

√8000= √400 √20= 20√20

 

√300= √100 √3= 10√3

 

√280= √4 √70 = 2√70