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QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

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App support, similar apps.  ## Get step-by-step solutions to your math problems ## Try Math Solver ## Get step-by-step explanations  ## Practice, practice, practice ## Get math help in your language ## Free Wolfram|Alpha Math Resources ## Wolfram|Alpha Website

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Question 1: To find the exact length of the perimeter of the triangle with side lengths 2√6 cm, 2√24 cm, and √54 cm, simply add the lengths of all three sides:

Perimeter = 2√6 + 2√24 + √54

Now, let's simplify each of these square roots: √6 = √(2 * 3) = √2 * √3 √24 = √(4 * 6) = 2√6 √54 = √(9 * 6) = 3√6

Now, plug these values into the perimeter formula:

Perimeter = 2(√2 * √3) + 2(2√6) + 3√6

Perimeter = 2√6 + 4√6 + 3√6

Perimeter = 9√6 cm

So, the exact length of the perimeter is 9√6 cm.

Question 2: (a) Rationalize the denominator for 6/∛(3y^2): To rationalize the denominator, multiply both the numerator and denominator by the cube root of the denominator:

6/∛(3y^2) * (∛(3y^2))^2 / (∛(3y^2))^2

This simplifies to:

(6∛(3y^2)) / 3y^2

Now, divide both the numerator and denominator by 3 to simplify:

(2∛(3y^2)) / y^2

(b) Rationalize the denominator for (√5 + 4) / (2 - √5): To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator:

[(√5 + 4) * (2 + √5)] / [(2 - √5) * (2 + √5)]

Now, simplify both the numerator and denominator:

Numerator: (2√5 + 8 + √5) Denominator: (4 - 5)

Combine like terms in the numerator:

(3√5 + 8) / (-1)

So, the rationalized expression is:

Question 3: (a) To solve for x in the equation x - 3√(x-5) = 5, first isolate the square root term on one side:

x - 5 = 3√(x-5)

Now, square both sides to eliminate the square root:

(x - 5)^2 = (3√(x-5))^2

x^2 - 10x + 25 = 9(x - 5)

Expand and simplify:

x^2 - 10x + 25 = 9x - 45

Now, move all terms to one side to set the equation to zero:

x^2 - 10x - 9x + 25 + 45 = 0

x^2 - 19x + 70 = 0

(x - 14)(x - 5) = 0

Now, set each factor equal to zero and solve for x:

x - 14 = 0 x = 14

x - 5 = 0 x = 5

Check both solutions, but the only valid solution is x = 5.

(b) To solve for x in the equation √(2x+6) = √(7-2x) + 1, first isolate one square root term on one side:

√(2x+6) - √(7-2x) = 1

Now, square both sides to eliminate the square roots:

(√(2x+6) - √(7-2x))^2 = 1^2

Simplify the left side:

2x + 6 - 2√((2x+6)(7-2x)) + 7 - 2x = 1

Now, simplify further:

2√((2x+6)(7-2x)) = 1 - 7 + 6

2√((2x+6)(7-2x)) = 0

Now, square both sides again to eliminate the square root:

4(2x+6)(7-2x) = 0

Now, expand and simplify:

4(14x - 4x^2 - 12x + 42) = 0

4(2x^2 - 26x + 42) = 0

Divide both sides by 4:

2x^2 - 26x + 42 = 0

x^2 - 13x + 21 = 0

(x - 7)(x - 3) = 0

x - 7 = 0 x = 7

x - 3 = 0 x = 3

Check both solutions, but the only valid solution is x = 7.

Question 4: The distance from home plate to second base on a baseball diamond is 90 feet. To find the distance the ball travels from home to second, it travels along the hypotenuse of a right triangle, with the bases forming the other two sides. Since the bases are 90 feet apart, this forms a right-angled triangle.

By the Pythagorean Theorem, the distance traveled by the ball is:

Distance = √(90^2 + 90^2) = √(8100 + 8100) = √16200

Now, simplify the square root:

Distance = √(900 * 18) = 30√18 feet

To approximate to two decimal places:

Distance ≈ 30 * 4.2426 ≈ 127.278 feet

So, the exact distance is 30√18 feet, and the approximate distance to two decimal places is approximately 127.28 feet.

Here's a simplified explanation of each of your questions in four steps:

Question 1: Finding the perimeter of a triangle.

• Add the lengths of all three sides.
• Simplify the square roots in the side lengths.
• Combine like terms.
• The exact perimeter is 9√6 cm.

Question 2: Rationalizing denominators.

• Multiply both the numerator and denominator by the cube root of the denominator to rationalize.
• Simplify by canceling common factors.
• The rationalized expression is obtained.
• Repeat the same process for the second part of the question.

Question 3: Solving equations with square roots.

• Isolate the square root term on one side of the equation.
• Square both sides to eliminate the square root.
• Simplify and move all terms to one side.
• Factor or solve the quadratic equation to find the values of x.

Question 4: Using the Pythagorean Theorem.

• Recognize that the bases and the ball's path form a right-angled triangle.
• Apply the Pythagorean Theorem: a^2 + b^2 = c^2.
• Substitute the given distances for the bases (90 feet).
• Simplify to find the exact and approximate distance the ball travels from home to second base.

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