Unit 8 Remediation

Simplifying Radicals

Remember...

However, some radicals are not perfect and we get decimals...

But...

Simplifying non-perfect radicals means "rescuing all the perfect squares you can from under the radical by factoring and leaving the rest behind...

Example... >

Then...

So...

Multiple Choice

Which is the prime factorization of $$\sqrt{20}$$  ?

• $$\sqrt{10\ +\ 10}$$  

• $$\sqrt{5\cdot2\cdot2}$$  

• $$\sqrt{10\cdot2}$$  

Steps to simplify

• Write it down

• Complete prime factorization

• Circle groups of 2 in the prime factorization

• Anything that was circled goes out of the radical and gets multiplied.  Anything not circled stays under the radical.  Example:  If you have circled two groups of 2 and a group of 3s then 2x2x3=12 goes out of the radical

• Check using Desmos for equivalency

Multiple Choice

Simplify $$\sqrt{384}$$  

• $$8\sqrt{6}$$  

• $$5\sqrt{5}$$  

• $$3\sqrt{7}$$  

• 10

Multiple Choice

Simplify $$\sqrt{288}$$  

• $$5\sqrt{7}$$  

• $$7\sqrt{7}$$  

• $$12\sqrt{2}$$  

• 16

Multiple Choice

Simplify $$\sqrt{54}$$  

• $$7\sqrt{3}$$  

• $$3\sqrt{7}$$  

• $$5\sqrt{5}$$  

• $$3\sqrt{6}$$  

Multiple Choice

Simplify $$\sqrt{80}$$  

• $$16\sqrt{2}$$  

• $$4\sqrt{5}$$  

• $$7\sqrt{5}$$  

• $$8\sqrt{2}$$  

Yo. Check it...

Multiple Choice

Factor $$\sqrt{x^5y^3}$$  

• $$\sqrt{x\cdot x\cdot x\cdot y\cdot y\cdot y}$$  

• $$\sqrt{x\cdot x\cdot x\cdot x\cdot x\cdot y}$$  

• $$\sqrt{x\cdot x\cdot x\cdot x\cdot x\cdot y\cdot y\cdot y}$$  

So how do we deal with something like...

Example...

factor and pull out the doubles...

Multiple Choice

Factor: $$\sqrt{x^3}$$  

• $$\sqrt{3x}$$  

• $$\sqrt{x\cdot x\cdot x}$$  

• $$\sqrt{x\cdot x}$$  

Multiple Choice

Simplify: $$\sqrt{x\cdot x\cdot x}$$  

• $$x\sqrt{x}$$  

• $$x^2\sqrt{x}$$  

• $$x\sqrt{x^2}$$  

Multiple Choice

Factor: $$\sqrt{8m^2n^4}$$  

• $$\sqrt{8\cdot m\cdot m\cdot n\cdot n\cdot n\cdot n}$$  

• $$\sqrt{4\cdot2\cdot m\cdot m\cdot n\cdot n\cdot n\cdot n}$$  

• $$\sqrt{2\cdot2\cdot2\cdot m\cdot m\cdot n\cdot n\cdot n\cdot n}$$  

Multiple Choice

Simplify: $$\sqrt{2\cdot2\cdot2\cdot m\cdot m\cdot n\cdot n\cdot n\cdot n}$$  

• $$2n^2\sqrt{2m^2}$$  

• $$2mn\sqrt{2n}$$  

• $$2mn^2\sqrt{2}$$  

Multiple Choice

Simplify: $$\sqrt{3x^6}$$  

• $$3x^3$$  

• $$x^3\sqrt{3}$$  

• $$x^2\sqrt{3x}$$  

Recall our calculator trick...

​Even Exponents

​*Divide the exponent by 2

Multiple Choice

Simplify: $$\sqrt[]{c^6}$$  

• $$c^2$$  

• $$c^3$$  

• $$c^4$$  

​What do you do if the exponent is odd?​

Recall our calculator trick...

Odd Exponents:

​*Divide the exponent by 2

If we get a decimal that means we have one left behind and the number in the front of the decimal tells us how many come out!

Multiple Choice

Simplify: $$\sqrt[]{e^9}$$  

• $$e^5\sqrt[]{e^4}$$  

• $$e^8\sqrt[]{e}$$  

• $$e^4\sqrt[]{e}$$  

Multiple Choice

Simplify: $$\sqrt[]{125c^9}$$  

• $$5c^4\ \sqrt[]{5c^5}$$  

• $$5c^4\ \sqrt[]{5c}$$  

• $$25c^4\ \sqrt[]{5c}$$  

Multiple Choice

Simplify: $$\sqrt[]{54c^6d^5}$$  

• $$3c^3d^2\ \sqrt[]{6d}$$  

• $$9c^3d^2\ \sqrt[]{6d}$$  

• $$6c^3d^2\ \sqrt[]{3d}$$  

Add/Subtract Radicals

Multiple Choice

Add:  4√7 - 10 √7 + 7√7

• -6√7
• 14√7
• -3√14
• √7

Multiple Choice

$$2\sqrt{27}+8\sqrt{3}+5\sqrt{8}$$  

• $$24\sqrt{3}$$  

• $$24\sqrt{3}+5\sqrt{8}$$  

• $$14\sqrt{3}+10\sqrt{2}$$  

Multiple Choice

$$\sqrt{12}+5\sqrt{18}+2\sqrt{27}$$  

• $$8\sqrt{3}+15\sqrt{2}$$  

• $$8\sqrt{2}+15\sqrt{3}$$  

• $$25\sqrt{2}$$  

Multiple Choice

Simplify:

$$\sqrt{50x^{12}y^5}$$  

• $$5x^6y^2\sqrt{2y}$$  

• $$2x^6y^2\sqrt{5y}$$  

• $$5y\sqrt{2x^6y^2}$$  

• $$5x^{12}y^5\sqrt{2y}$$  

Multiple Choice

I can simplify radical expressions that include addition and subtraction.

Simplify the radical expression below.

$$\sqrt{175}-\sqrt{28}=$$  

• $$21\sqrt{7}$$  

• $$3\sqrt{7}$$  

• $$7\sqrt{7}$$  

Multiple Choice

$$2+2\sqrt[]{2}+\sqrt[]{8}+3\sqrt[]{2}$$  

• $$2+7\sqrt[]{2}$$  

• $$2+\sqrt[]{8}+7\sqrt[]{2}$$  

• $$2+7\sqrt[]{10}$$  

• $$\sqrt[]{8}+9\sqrt[]{2}$$  

Multiple Choice

6√2 ⋅5√14

• 32√7
• 60√7
• 2√7
• 30√7

Multiple Choice

$$5\sqrt{2}\cdot4\sqrt{10}$$  

• $$20\sqrt{20}$$  

• $$10\sqrt{40}$$  

• $$40\sqrt{5}$$  

• $$9\sqrt{13}$$  

Multiple Choice

-2√10 ⋅ 3√20

• -60√2
• 10√2
• √30
• -6√200

Multiple Choice

Find the area of the rectangle shown.

• 18√20
• 12√10 + 6√2
• 9√12
• 36√5