Conic sections #3 - Questions and Answers

Question 1 The rational exponent form of a⁰.⁴ in radical form. Answer 1 Step 1 Convert the given expression a⁰.⁴ in the radical form as follows: a⁰.⁴ = a^(4/10) = a^(2/5) = (a²)^(1/5) = ⁵√a² Thus, the rational exponent form of a⁰.⁴ in radical form is ⁵√a² Answer 2 Change 0.4 into a fraction. Multiply by 10/10 to remove the decimal. 10 * 0.4 / 10 = 4/10 a^(4/10) Multiply 10 by 0.4. a^(4/10) Cancel the common factor of 4 and 10 Factor 2 out of 4. a^(2(2)/10) Cancel the common factors. Factor 2 out of 10. a^(2(2)/ 2(5)) Cancel the common factor. a^(2/5) Rewrite the expression. a^(2/5) Apply the rule x^(m/n) = ⁿ√x^m to rewrite the exponentiation as a radical. ⁵√a² Question 2 To determine: The radical form of ∛64 in the rational exponent form. Answer 1 Step 1 Given: The number is ∛64 Calculation: Convert the given number ∛64 in the rational exponent form as follows: ∛64 = 64^(1/3) = (4³)^(1/3) Thus, the radical form of ∛64 in the rational exponent form is (64)^(1/3) Answer 2 ∛64 Rewrite 64 as 4³ ∛4³ Pull terms out from under the radical. 4 Question 3 To convert the given radical expression √5² to its rational exponent form and then simplify Answer 1 Step 1 The given expression can be written as: √5² = (5²)^(1/4) (Since √a = a^(1/2)) √5² = (5^(2*1/4)) (Since (a^m)^n = a^(mn)) √5² = (5^(1/2)) √5² = √5 Therefore, the given radical expression to its rational exponent form is given by √5² = (5²)^(1/4) and the simplified form is given by √5²=√5 Answer 2 Step 1 The given expression can be written as: √5² = (5²)^(1/4) (Since √a = a^(1/2)) √5² = (5^(2*1/4)) (Since (a^m)^n = a^(mn)) √5² = (5^(1/2)) √5² = √5 Therefore, the given radical expression to its rational exponent form is given by √5² = (5²)^(1/4) and the simplified form is given by √5² = √5 Question 4 Derivatives of functions with rational exponents Find dy/dx y = (5x + 1)^(2/3) Answer Step 1 We have to find the derivative (dy/dx) of the function. y = (5x + 1)^(2/3) We know the formulas of derivatives, d/dx (x^n) = n x^(n-1) d/dx c = 0 (where c is some constant) d/dx f(g(x)) = n f(x)^(n-1) * d/dx (f(x)) (chain rule) Step 2 Differentiating the given function with respect to 'x' and applying above formula, we get dy/dx = d/dx( (5x+1)^(2/3) ) dy/dx = (2/3)(5x+1)^((2/3)-1) d/dx(5x+1) dy/dx = (2/3)(5x+1)^(-1/3) * (5*1+0) dy/dx = 2 * 5 / (3(5x+1)^(1/3)) dy/dx = 10 / (3(5x+1)^(1/3)) dy/dx= 10/( 3∛(5x+1)) Hence, derivative of the function is 10/(3∛(5x+1)) Question 5 What is an ellipse. Answer We know that a conic sections are curves that result from the intersection of a right circular cone and a plane. There are four conic sections: the circle, the ellipse, the parabola and the hyperbola. An ellipse is the set of all points P in a plane the sum of whose distances from two fixed. Points F₁ and F₂ is constant. Those two fixed points are known as foci of the ellipse. And the midpoint of two foci is the center of the ellipse. Line-segment joining the vertices of the ellipse is known as its major axis and the line segment perpendicular to the major axis and passing through center is known as its minor axis. The standard form of the equation of an ellipse with center at origin, major axis of length 2a and minor axis of length 2b is x²/a² + y²/b² = 1 or x²/b² + y²/a² = 1 For the equation x²/a² + y²/b² = 1, the major axis of the ellipse will be horizontal and for the equation x²/b² + y²/a² = 1, the major axis of the ellipse will be vertical. Compare the provided equation x²/16 + (y²/4) = 1 with the standard form x²/a² + y²/b² = 1, a² = 16 ⇒ a = 4 b² = 4 ⇒ b = 2