What is the simplest form of √8a^3? What is the rational form of 1/3? What is the simplest form of 7/18? Which of the following is the simplest form of √16a^5b^3? What is the simpl... What is the simplest form of √8a^3? What is the rational form of 1/3? What is the simplest form of 7/18? Which of the following is the simplest form of √16a^5b^3? What is the simplest form of √75? Evaluate 2+√10/√5. What is the conjugate of -2/√5? What is the sum of 2√3 + 4√3? Evaluate 3√2 + 5√3 - 3√3. Which of the following statements is not true? What is the value of x on the equation √x = 49? What is the first step to do to solve √x + 2 = 6? Which of the following is the value of x on number 34? Given √2x + 3 = √x + 6, what is x? Which of the following equations will you need to solve the problem? What is the distance between the airplane and the helicopter?
Understand the Problem
The question is asking for various mathematical simplifications and evaluations, including identifying the simplest forms of expressions involving square roots and the operations on radicals. Additionally, it addresses a problem involving the distance between two objects, applying mathematical concepts possibly related to the Pythagorean theorem.
Answer
The distance between the airplane and the helicopter is \( 10 \) miles.
Answer for screen readers
The distance between the airplane and the helicopter is ( 10 ) miles.
Steps to Solve
- Identifying the Problem Type
The problem consists of various mathematical questions, primarily involving simplification of radicals and distance calculations using the Pythagorean theorem.
- Simplifying Radicals
For each algebraic expression, simplify where necessary. For example, for ( \sqrt{8a^3} ):
- Factor out the perfect squares: ( \sqrt{(4)(2)(a^2)(a)} = \sqrt{4} \cdot \sqrt{2} \cdot a = 2a\sqrt{2} ).
- Finding Rational Forms
To find the rational form of ( \frac{1}{\sqrt{3}} ):
- Multiply numerator and denominator by ( \sqrt{3} ): $$ \frac{\sqrt{3}}{3} $$
- Pythagorean Theorem for Distance
Given that a plane is 8 miles north and a helicopter is 6 miles east:
- The distance ( d ) between them can be calculated using: $$ d = \sqrt{(8^2 + 6^2)} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ miles} $$
- Evaluating Specific Problems in the List
Each problem such as simplifying ( \sqrt{7/18} ) or finding conjugates follows the same principle of identifying common factors or appropriate operations.
The distance between the airplane and the helicopter is ( 10 ) miles.
More Information
This problem illustrates the application of the Pythagorean theorem, which is essential in finding distances in a right triangle formed by the distances traveled north and east. This theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Tips
- Confusing the operations on radicals, such as adding instead of multiplying when dealing with common radicands.
- Incorrectly applying the Pythagorean theorem to find the distance, often forgetting to square the sides or miscalculating square roots.
- Neglecting to rationalize denominators when working with fractions involving radicals.
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