Simplify by rationalizing each denominator.
1. 2/√3
A: 2√3 / 3
2. -18/√6
A: -3√6
Add or subtract.
3. 4√3 - 9√3
A: -5√3
4. √8 - 15√2
A: -13√2
5. √45 + √20
A: 5√5
6. 2√48 + 2√12
A: 12√3
Simplify each expression.
7. √162
A: 9√2
8. √50/9
A: (5√2)/ (3)
9. (√288) / (√8)
A: 6
10. (2√126) / (√14)
A: 6
11. The largest mosaic in the world is on the walls of the central library of the Universidad Nocional Autónoma de México in Mexico City. The mosaic depicts scenes from the nation's history and covers an area of 4000 m^2. If the entire mosaic were on one square wall, what would its dimensions be?
A: The dimensions would be 20√10 m^2 by 20√10 m^2.
1. 2/√3
A: 2√3 / 3 yes
2. -18/√6
A: -3√6 yes
Add or subtract.
3. 4√3 - 9√3
A: -5√3 yes
4. √8 - 15√2
A: -13√2 yes
5. √45 + √20
A: 5√5 yes
6. 2√48 + 2√12
= 2 sqrt (4*12) + 2 sqrt 12
= 4 sqrt 12 + 2 sqrt 12
= 6 sqrt 12
= 12 sqrt 3
A: 12√3 so yes
Simplify each expression.
7. √162
A: 9√2 yes
8. √50/9
A: (5√2)/ (3) I guess so if it is really sqrt (50/9) and not (1/9) sqrt 50 as you wrote it
9. (√288) / (√8)
A: 6 yes
10. (2√126) / (√14)
A: 6 yes
What about #11?
11 yes
Thank you!
To simplify the expressions by rationalizing the denominator, you need to eliminate any square roots in the denominator by multiplying both the numerator and denominator by an appropriate form of 1.
1. For 2/√3, multiply both the numerator and denominator by √3 to get (2 * √3) / (√3 * √3) = 2√3 / 3.
2. For -18/√6, multiply both the numerator and denominator by √6 to get (-18 * √6) / (√6 * √6) = -18√6 / 6. Simplify further by dividing both the numerator and denominator by 6 to get -3√6.
To add or subtract expressions with square roots, you can only combine like terms. Terms that have the same radical part can be added or subtracted.
3. For 4√3 - 9√3, both terms have the same radical part, √3. Subtract the coefficients (4 - 9 = -5) and keep the radical part unchanged: -5√3.
4. For √8 - 15√2, there are no like terms to combine.
5. For √45 + √20, there are no like terms to combine.
6. For 2√48 + 2√12, there are no like terms to combine.
To simplify square roots, look for perfect square factors under the radical symbol and simplify them.
7. For √162, break down 162 into its perfect square factors: √(2 * 3^4) = √(2 * 9^2) = √(2 * (3^2)^2) = √(2 * 3^2)^2 = 3 * √2 = 3√2.
8. For √50/9, simplify the square root: √(5^2 * 2) = 5√2. Divide that by 9 to get (5√2) / 3.
9. For (√288) / (√8), simplify the square roots: √(2^5 * 3^2) / √(2^3) = 2^2 * 3 / 2 = 4 * 3 / 2 = 6.
10. For (2√126) / (√14), simplify the square roots: (√(7 * 2^2 * 3)) / (√(2 * 7)) = 2 * √3 = 6.
For question 11, you are asked to find the dimensions of a square wall that would be equivalent to the given area of 4000 m^2. To find the length of one side of the square, you can take the square root of the area.
11. (√4000) = √(2^4 * 5^2 * 10^2) = 2^2 * 5 * 10 = 20√10.
Therefore, the dimensions of the square wall would be 20√10 m by 20√10 m.