Excelsior Algebra I Unit 9

Simplify the expression: (√2 + √10) / (√18 + √10)

The answer is (-1 - √5) / 2 To solve this, you will need to simplify anything that you can first. Then, you will need to rationalize the denominator by using the conjugate. (√2 + √10) / (√18 + √10) Only the √18 can be broken down. √18 = √9 • 2 = √9 • √2 = 3√2 (√2 + √10) / (3√2 + √10) = Rationalize by multiplying top and bottom by the denominator's conjugate. (√2 + √10) / (3√2 + √10) • (3√2 - √10) / (3√2 - √10) = Foil the top and then foil the bottom (3√4 - √20 + 3√20 - √100) / (9√4 - 3√20 + 3√20 - √100) = Combine like terms in the top and then the bottom (3 • 2 - 2√20 - 10) / (9 • 2 - 10) = Simplify (6 - 2√20 - 10) / (18 - 10) = Simplify ( -4 - 2√20) / 8 = Simplify the √20 = √4 • 5 = √4 • √5 = 2√5 (-4 -4√5) / 8 = Reduce (-1 - √5) / 2

Simplify the expression: 7 / (√5 - √2)

The answer is (7√5 + 7√2) / 3 One of the rules that you cannot break is that you cannot have radicals in denominators. In order to remove them, you need to multiply the numerator and the denominator (top and bottom) by the conjugate of the bottom radical. The conjugate means to change the sign. 7 / (√5 - √2) • (√5 + √2) / (√5 + √2) = 7(√5 + √2) / (√5 - √2)(√5 + √2) = Distribute the top and FOIL the bottom 7√5 + 7√2 / √25 - √10 + √10 - √4 = The middle terms on the bottom cancel each other out 7√5 + 7√2 / √25 - √4 = Find the square roots if possible (7√5 + 7√2) / 5 - 2 = (7√5 + 7√2) / 3

Simplify the radical expression: - 5 √90

The answer is -15 √10 To solve this, you have to break down the radicand (the stuff behind the radical) into perfect squares. In this case, the 90 is not a perfect square. You must break all non-perfect-squares into perfect square factors, if there are any. For the 90, the perfect square factors are 9 and 10. The square root of 9 is 3. The outside of the radical starts with a -5 outside, and we took the square root of 9 to get 3 which will be multiplied all together as the following: -5(3) = -15 The 10 is the only factor that cannot be broken down any further and it remains underneath the radical.

Multiply: (3 / 4x) • (6 / 2x^2)

The answer is 9 / 4x^3 To solve, you can cross-cancel first if possible, or you can multiply across the top and bottom separately and then cross-cancel anything you can. (3 / 4x) • (6 / 2x^2) = 18 / 8x^3 = Reduce by 2 9 / 4x^3

Solve the equation. Check your solution. √(x+4) = 12

The answer is 140 To solve, get the variable, x, by itself. Right now, the x is with a plus four underneath the radical. To get rid of the radical, you have to undo the square root with squaring and square both sides. To get rid of the plus four, you have to undo the addition with subtraction and subtract 4 from both sides. √(x+4) = 12 √(x+4)^2 = 12^2 square both sides x + 4 = 144 x + 4 - 4 = 144 - 4 Subtract 4 from both sides x = 140

Simplify the expression: (5 - √10)(5 + √10)

The answer is 15 To solve this, you must FOIL. When you multiply the FIRST terms: 5 • 5 = 25 When you multiply the OUTSIDE terms: 5 • √10 = 5√10 When you multiply the INSIDE terms: 5 • (-√10) = -5√10 When you multiply the LAST terms: (-√10)(√10) = -√100 = -10 25 + 5√10 - 5√10 - 10 = 25 - 10 = 15

A scuba diver has a taut rope connecting the dive boat to an anchor on the ocean floor. The rope is 180 feet long and the water is 60 feet deep. To the nearest tenth of a foot, how far is the anchor from a point directly below the boat?

The answer is 169.7 ft. To solve this, you will need to use the Pythagorean Theorem, which is a^2 + b^2 = c^2. The water is 60 feet deep, so a = 60 feet. The anchor line is 180 feet and creates the hypotenuse, so c = 180. Substitute in the values given. Remember that the c is always the hypotenuse which is across from the right angle in a right triangle. 60^2 + b^2 = 180^2 3600 + b^2 = 32400 b^2 = 32400 - 3600 b^2 = 28800 Take the square root b = 169.7 feet

Find the length of the missing side of the triangle. If necessary, round to the nearest tenth. a = 8, b = 15 , and c = ??

The answer is 17 To solve this, you will need to use the Pythagorean Theorem, which is a^2 + b^2 = c^2. Substitute in the values given. Remember that the c is always the hypotenuse which is across from the right angle in a right triangle. 8^2 + 15^2 = c^2 64 + 225 = c^2 289 = c^2 Take the square root 17 = c

Simplify the expression: 3√6 + 9√54

The answer is 30√6 In order to combine radicals, they have to be like radicals. Since 6 and 54 are not the same, they cannot by combined yet. To solve this, break down each radicand. The 6 cannot be broken down into any perfect square factors, but the 54 can. The 54 can be broken into 9 and 6. The square root of the 9 is 3 and will be removed from under the radical and the 6 will remain underneath. 3√6 + 9√54 = 3√6 + 9•3√6 = 3√6 + 27√6 From this point, you will add the numbers in front of the like radicals together. So 3 + 27 = 30 and the √6 gets added to the end.

Simplify the radical expression: √(72 x^17 y^7 / 8x y^13)

The answer is 3x^8 / y^3 To solve this, you have to break down the radicand (the stuff behind the radical) into perfect squares. In this case, none of the parts are perfect squares. You must break all non-perfect-squares into perfect square factors, if there are any. Since this is a division problem, you can divide the numbers first if you want. 72/ 8 = 9 which is a perfect square. x^17 / x = x^16 which is a perfect square. y^7 / y^13 = 1 / y^6 which is a perfect square. The square root of 9 is 3, the square root of x^16 is x^8, and the square root of y^6 is y^3. The outside of the radical starts with nothing outside, so there is nothing to multiply. There are no non-perfect-square factors that cannot be broken down any further underneath the radical.

Simplify the radical expression by rationalizing the denominator. 8√(100) / √(600)

The answer is 4√(6) / 3 To solve this, break down each radicand first. 100 is a perfect square with its square root being 10. 600 is not a perfect square, but it can break into 100 and 6. The square root of 100 is 10. 8•10 / 10√6 = 80 / 10√6 One of the rules that you cannot break is that you cannot have radicals in denominators. In order to remove them, you need to multiply the numerator and the denominator (top and bottom) by the bottom radical. 80 (√6) / (10√6)(√6) = 80√6 / 10 • 6 = 80 √6 / 60 = (8√6) / 6 = 4√6 / 3

Solve the equation. Check your solution. √(3x + 6) = √(6x - 9)

The answer is 5 To solve, you must get the variable, x, by itself. Right now, there are two x's- one on each side of the equal sign. To get rid of the radicals, you have to undo the square root with squaring both sides. Next you gather the x terms to one side and then get x by itself. √(3x + 6) = √(6x - 9) √(3x + 6)^2 = √(6x - 9)^2 Square both sides 3x + 6 = 6x - 9 Gather the x terms to one side by subtracting 3x from both sides 3x + 6 - 3x = 6x - 9 - 3x 6 = 3x - 9 Add 9 to both sides 6 + 9 = 3x - 9 + 9 15 = 3x Divide each side by 3 15 / 3 = 3x / 3 5 = x

Simplify the radical expression: √(432 / 6)

The answer is 6 √2 To solve this, you have to break down the radicand (the stuff behind the radical) into perfect squares. In this case, neither the 432 nor the 6 is a perfect square. You must break all non-perfect-squares into perfect square factors, if there are any. Since this is a division problem, you can divide the numbers first if you want. 432 / 6 = 72. For the 72, the perfect square factors are 36 and 2. The square root of 36 is 6. The outside of the radical starts with nothing outside, so there is nothing to multiply. The 2 is the only factor that cannot be broken down any further and it remains underneath the radical.

Find the length of the missing side of the triangle. If necessary, round to the nearest tenth. a = ??, b = 24 , and c = 25

The answer is 7 To solve this, you will need to use the Pythagorean Theorem, which is a^2 + b^2 = c^2. Substitute in the values given. Remember that the c is always the hypotenuse which is across from the right angle in a right triangle. a^2 + 24^2 = 25^2 a^2 + 576 = 625 a^2 = 625 - 576 a^2 = 49 Take the square root a = 7

Simplify the rational expression. 8y^2 / (y^3 - y^2)

The answer is 8 / y - 1 To solve, you must factor the top and bottom separately and then cross-cancel anything you can. 8y^2 cannot be factored (y^3 - y^2) factors to y^2 (y - 1) 8y^2 / y^2 (y - 1) The y^2 get cross-cancelled 8 / y- 1

Multiply: [(x^2 - 9) / 5x] • [(8x) / (x + 3)]

The answer is 8(x - 3) / 5 To solve, you must factor the top and bottom separately and then cross-cancel anything you can. x^2 - 9 factors to (x + 3)(x - 3) Nothing else is factorable (x + 3)(x - 3) / 5x • 8x / (x + 3) = Cross-cancel the x and the (x + 3) (x - 3) / 5 • 8 / 1 = Multiply 8(x - 3) / 5

Solve the equation. Check your solution. 3 = √(x) - 6

The answer is 81 To solve, get the variable, x, by itself. Right now, the x is with a negative 6 and a square root. To get rid of the negative 6, you have to undo the subtraction with addition and add 6 to both sides. To get rid of the square root, you have to undo the square root with squaring and square both sides. 3 = √(x) - 6 3 + 6 = √(x) - 6 + 6 Add 6 to both sides 9 = √(x) 9^2 = √(x^2) Square both sides 81 = x

Simplify the radical expression: √(96 w^5 / 16)

The answer is w^2√(6w) To solve this, you have to break down the radicand (the stuff behind the radical) into perfect squares. In this case, only the 16 is already a perfect square. The 96 and the w^5 are not perfect squares. You must break all non-perfect-squares into perfect square factors, if there are any. For the 96, the perfect square factors are 16 and 6 and for the w^5, the perfect square factors are w^4 and w. The square root of 16 is 4. The outside of the radical starts with nothing outside, and we took the square root of 16 and w^4 in the numerator, 16 in the denominator. The square roots are 4 and w in the numerator and 4 in the denominator which will be multiplied all together as the following: (4)(w^2) / 4 = 4 w^2 / 4 The 6 and the w are the only factors that cannot be broken down any further and it remains underneath the radical.

Simplify the rational expression. 4x + 12 / 4x - 12

The answer is x + 3 / x - 3 To solve, you must factor the top and bottom separately and then cross-cancel anything you can. 4x + 12 factors to 4(x + 3) 4x - 12 factors to 4(x - 3) 4(x + 3) / 4 (x - 3) The fours get cross-cancelled x + 3 / x - 3

Divide: x^2 - 16 / x - 8 ÷ (x - 4)

The answer is x + 4 / x - 8 To solve, you must factor the top and bottom separately and then cross-cancel anything you can. You must also remember that you cannot divide fractions, but instead flip over the second fraction and multiply. x^2 - 16 factors to (x + 4) (x - 4) Nothing else is factorable (x + 4)(x - 4) / (x - 8) ÷ 1 / (x - 4) = Cross-cancel the x - 4 (x + 4) / (x - 8)

Simplify the rational expression. (x^2 + 4x - 32) / (x^2 + x - 56)

The answer is x - 4 / x - 7 To solve, you must factor the top and bottom separately and then cross-cancel anything you can. (x^2 + 4x - 32) factors to (x + 8)(x -4) (x^2 + x - 56) factors to (x + 8)(x - 7) (x + 8)(x -4) / (x + 8)(x - 7) The x + 8 gets cross-cancelled x - 4 / x - 7

Simplify the rational expression. (x^2 - x - 20) / (x^2 - 4x - 32)

The answer is x - 5 / x - 8 To solve, you must factor the top and bottom separately and then cross-cancel anything you can. (x^2 - x - 20) factors to (x + 4)(x - 5) (x^2 - 4x - 32) factors to (x + 4)(x - 8) (x + 4)(x - 5) / (x + 4)(x - 8) The x + 4 gets cross-cancelled x - 5 / x - 8

Divide: [(x^2 - 4x) / (x^2 + 2x - 8)] ÷ [(x - 4) / (x + 4)]

The answer is x / x - 2 To solve, you must factor the top and bottom separately and then cross-cancel anything you can. You must also remember that you cannot divide fractions, but instead flip over the second fraction and multiply. x^2 - 4x factors to x (x - 4) x^2 + 2x - 8 factors to (x - 2)(x + 4) [x (x - 4) / (x - 2)(x + 4)] ÷ [(x + 4) / (x - 4)] Cross-cancel the x - 4 and the x + 4 x / x - 2

Simplify the radical expression: √(5 / 64 p^6)

The answer is √(5) / 8p^3 To solve this, you have to break down the radicand (the stuff behind the radical) into perfect squares. In this case, the 64 and the p^6 are already perfect squares. The 5 is not a perfect square. You must break all non-perfect-squares into perfect square factors, if there are any. For the 5, there are no perfect square factors. The square root of 64 is 8, and the square root of p^6 is p^3. The outside of the radical starts with nothing outside, and we took the square roots of 64, and p^6 in the denominator as 8, and p^3 which will be multiplied all together as the following: (8)(p^3) = 8 p^3 The 5 is the only factor that cannot be broken down any further and it remains underneath the radical.

Simplify the radical expression: -3 √(250 x^8 y^4)

The answer is: -15 x^4 y^2 √10 To solve this, you have to break down the radicand (the stuff behind the radical) into perfect squares. In this case, the x^8 and the y^4 are already perfect squares. The 250 is not a perfect square. You must break all non-perfect-squares into perfect square factors, if there are any. For the 250, the perfect square factors are 25 and 10. The square root of 25 is 5 and the 10 will remain underneath the radical. The square root of x^8 is x^4 and the square root of y^4 is y^2. The outside of the radical starts with a -3, and we took the square roots of 25, x^8, and y^4 as 5, x^4, and y^2 which will be multiplied all together as the following: (-3)(5)(x^4)(y^2) = -15 x^4 y^2 The 10 is the only factor that cannot be broken down any further and it remains underneath the radical.