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The square root of 19 squared plus 19 plus 20 is calculated to be 20.

To find (a+b)^2, given a^2 + b^2 = 29 and ab = 10, substitute the given values into the equation (a+b)^2 = a^2 + 2ab + b^2, resulting in (a+b)^2 = 49.

The expression 3³ ÷ 3 ÷ 3 simplifies to 1.

The only real solution to the cubic equation a^3 + a^2 - 36 = 0 is a = 3.

The solution to the equation X^3 - X^0 = 26 is X = 3.

24 is 60 percent of 40.

The determinant of the given 4x4 matrix is calculated using cofactor expansion along the fourth row, resulting in a determinant of -21.

The determinant of the 2x2 matrix [[2, 2], [3, 5]] is calculated as (2 * 5) - (2 * 3) = 4.

The equation (x-1)^(|x|-4) = 1 has solutions x = -4, 0, 2, and 4.

The difference of squares formula is used to evaluate the expression 100^2 - 99^2, resulting in 199.

To find the smallest possible values for A and B in a 10-digit number divisible by both 8 and 9, the last three digits (9B8) must be divisible by 8, and the sum of all digits must be divisible by 9. This leads to B=2 and A=3 as the smallest possible values.

The equation x^2 - 2x = 0 has two solutions, x = 0 and x = 2, not just x = 2.

The solution for b in the equation 2^(b^2+2) = 64 is b = 2.

To find 2^(m+n), solve for m and n in the equations 2^m = 1 and 2^n = 16, then substitute the values into the expression.

The expression 0^0 is considered an indeterminate form in calculus because different approaches to evaluating it yield different potential values. While some approaches suggest a value of 1 and others suggest 0, the limit does not have a single, well-defined value. In some contexts, like combinatorics, it is defined as 1 for convenience.

The magnitude of the given row vector is calculated, resulting in 3.

The determinant of the 3x3 matrix is calculated using the formula a(ei - fh) - b(di - fg) + c(dh - eg), resulting in a determinant of 0.

The multiplication of the two given matrices results in a 2x3 matrix with entries calculated by multiplying corresponding rows and columns of the matrices and summing the products.

The provided document details the simplification of algebraic fractions and the solution of algebraic expressions. Part 1 simplifies various rational expressions by factoring the numerators and denominators, canceling common factors. Part 2 involves adding, subtracting, and multiplying rational expressions, often requiring finding common denominators and simplifying the resulting expressions.

The equation 3^x - 2^x = 65 has a solution at x = 4.

The equation x^x + 6/x = 7 has two solutions, x = 1 and x = 2, as demonstrated by direct substitution.

The unknown angle is 50 degrees.

The solution to the equation 3^(x+2) = 2^(x+3) is found by taking the natural logarithm of both sides, applying logarithm properties, isolating x, and simplifying to an expression involving natural logarithms of 2 and 3. The approximate value of x is then calculated.

The expression $((i+1)(i-1))^2$ simplifies to 4.

The solution to the problem \sqrt{25\%} + \sqrt{49\%} is 1.2, which is equivalent to 120%.

To find x, the exterior angle theorem is used, along with the knowledge that a right angle is 90 degrees, to solve for x, which equals 45 degrees.

The solution to the equation involving radicals and exponents is x = 77/120.

The water lily takes 47 days to cover half the pond, given that it doubles in size each day and covers the entire pond on day 48.

The equation 3^(x^2)/9^x = 81 is solved by rewriting the equation with the same base, simplifying the denominator, applying the quotient rule for exponents, equating the exponents, rearranging into a quadratic equation, and finally using the quadratic formula to find the two solutions, x = 1 + √5 and x = 1 - √5.

The unknown angle in a triangle, formed by two slanted lines and a vertical line, with two given angles of 80 degrees each, is calculated to be 20 degrees using the triangle angle sum property.