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Simplify the nested radical expression √(5 + √11). Using a method involving squaring and the quadratic formula, it's determined that √(5 + √11) cannot be simplified further and remains in its simplest form.

If a/b = 0.25, find the value of 2a + 3b in terms of b. The solution shows step-by-step how to substitute and simplify the expression to get the final answer: 7b/2.

Express sums of Fibonacci numbers in terms of Fibonacci numbers. Find the simplified expressions for sums like F1+F2+...+F15, F1+F3+...+F29, and others, using Fibonacci identities. Solutions include F17-1, F30, and some cases with no simple closed form.

Solve the quadratic equation x² = (20-x)*10/2. The solutions are x = (-5 ± 5√17)/2, approximately x ≈ 8.09 or x ≈ -13.09.

This pre-calculus activity covers identifying conic sections (hyperbola, parabola, circle), proving trigonometric identities (sin(-θ) = -sin θ, csc²θ - cot²θ = 1), and evaluating functions (f(x) = 3x², g(x) = -2x + 7). Solutions are provided step-by-step for each problem.

Simple math problem: 1 + 1 equals 2. Find the solution and understand basic addition with this quick example.

Discover the pattern in the math equations 2^6 x 2^6 = 2^11 + 2^11 and 3^8 x 3^8 = 3^15 + 3^15 + 3^15. Learn how to generalize this pattern and create more examples.

This problem asks to find the value of x^x, given that x = 4^-2. The solution involves simplifying x, applying exponent properties, and calculating the final result, which is 1/√[4]2 or approximately 0.8409.

This math problem involves finding the sum of two angles (x and y) in a triangle. Using the fact that the angles in a triangle add up to 180 degrees and that angles on a straight line add up to 180 degrees, we can solve for x and y. The solution shows that x + y = 118 degrees.

This problem asks to solve for x in the equation 4^(2x-2) * 4^(-4x+1) = 1/16. The solution is x = 1/2.

This math problem asks to solve for A in the equation 2A⁻¹ + 1/2 = 1. The solution is A = 4.

This math problem involves finding the values of x and y in a triangle with angles expressed as algebraic expressions. The solution involves setting up equations based on the angle sum property of triangles and straight lines, and solving for x and y. The solution is x = 30 and y = 100.

This math problem asks to find the value of 'a' given that f(x) = (x-3)² and f(a) = a². The solution involves substituting 'a' into the function, expanding the equation, and solving for 'a'. The answer is a = 3/2 or 1.5.

This math problem involves solving for the ratio A/P given the equation (A+P)/(A-P) = 9/2. The solution involves cross-multiplication, rearranging terms, and expressing A in terms of P to find the value of A/P, which is 11/7.

This problem asks to find the area of a right-angled triangle inscribed in a circle, where the base of the triangle is the diameter of the circle. The diameter is given as 12 cm. The solution uses the formula for the area of a triangle and the fact that the height of the triangle is equal to the radius of the circle. The final answer is 36 cm².

Learn how to solve the definite integral of 1/x * (x/(1-x)) from 0 to 1. This step-by-step guide demonstrates simplifying the integrand, applying integration rules, and evaluating the limits to reveal the integral diverges to positive infinity.

A bonus of Rs 21000 is shared between 3 managers and 7 salespeople. A salesperson claims they would get 25% more if the bonus was shared equally. Is the salesperson correct? The answer is no, they would get about 16.67% more.

This math problem involves solving for the value of 3^X - 3^Y, given that 3^X ÷ 3^Y = 3 and 3^X + 3^Y = 12. The solution involves using the properties of exponents to simplify the equations and solve for the values of 3^X and 3^Y, ultimately leading to the answer of 6.

This math problem asks you to add four mixed numbers: 15 2/7 + 23 3/4 + 7 5/7 + 5 1/4. The solution involves finding a common denominator, converting the mixed numbers to improper fractions, adding the fractions, and simplifying the result. The answer is 52.

This problem involves solving for the value of P in the equation 125^(3P-4) = √5. The solution involves simplifying both sides of the equation, expressing them with the same base, and then equating the exponents to solve for P. The final answer is P = 25/18.

This math problem involves simplifying the expression 8^(-1/2) / √2. The solution involves using exponent rules and simplifying square roots to arrive at the final answer of 1/4 or 0.25.

This step-by-step solution demonstrates how to solve the equation 7√(3x - 5) = 28, arriving at the answer x = 7.

This math problem asks to solve for the value of T² + M, given the equations T + M = 6 and T - M = 2. The solution involves solving for T and M using the first two equations and then substituting the values into the third equation to find the answer: T² + M = 18.

This problem solves the equation 2^(x+1) - 3^x = 0 for x using logarithmic properties. The solution is x = 1 / (log₂(3) - 1), which is approximately 2.7095.

Solve the math problem 13 - 2 × 3! and find the answer. The solution is 1.

Solve for the value of √x + 1/√x given that x = 7 + 4√3. The solution reveals a surprising simplification, resulting in an answer of 4.

Solve the math problem 5 + 1 x 5. The correct answer is 10, following the order of operations.

Solve the math problem 60 ÷ 6 x 2. The answer is 20.

Solve for 'c' in the equation f = (a-c)(b-d)/(ac-bd). The solution is c = (ab - ad - fbd)/(fa + b - d).

This proof demonstrates that Bessel functions of different orders, $J_n(x)$ and $J_{n+m}(x)$, have no common zeros except at x=0. The proof uses the Bessel differential equation, recurrence relations, and a proof by contradiction to show that assuming a common zero leads to a contradiction.