What are the definitions of additive identity, mantissa, 3rd quadrant, medians, horizontal line, quadratic equation, triangle inequality theorem, and binomial cube expansion?

Noor, let's knock this out. 1. The additive identity in the set of real numbers (R) is the number that, when added to any real number, leaves the number unchanged. This number is 0. A) 0 2. In a logarithm, the mantissa is the fractional part of the logarithm. The integer part is called the characteristic. B) Mantissa 3. In the Cartesian coordinate system, a point with both negative x and negative y coordinates (e.g., (-4, -3)) is located in the 3rd quadrant. C) 3rd quadrant 4. The medians of a triangle are line segments joining a vertex to the midpoint of the opposite side. They intersect at the centroid, which divides each median in a 2:1 ratio. D) Medians 5. A horizontal line of the form y = c (where c is a constant, like y=5) is always parallel to the x-axis. A) x-axis 6. A quadratic equation is a polynomial equation where the highest power of the variable is 2. C) 2 7. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. B) Greater 8. Step 1: Recognize the expression as the expansion of a binomial cube. The formula for (a-b)^3 is a^3 - 3a^2b + 3ab^2 - b^3. Step 2: Expand the given expression. a^3 - 3ab(a-b) - b^3 = a^3 - (3a^2b - 3ab^2) - b^3 = a^3 - 3a^2b + 3ab^2 - b^3 Step 3: Compare with the formula. The expanded form matches (a-b)^3. B) (a-b)^3 9. Step 1: Use the distance formula between two points (x_1, y_1) and (x_2, y_2): D = sqrt((x_2-x_1)^2 + (y_2-y_1)^2) Step 2: Substitute the given points A(3,0) and B(0,3). D = sqrt((0-3)^2 + (3-0)^2) D = sqrt((-3)^2 + (3)^2) D = sqrt(9 + 9) D = sqrt(18) Step 3: Simplify the radical. D = sqrt(9 × 2) D = 3sqrt(2) The distance is 3sqrt(2) units. C) 3sqrt(2) units 10. Step 1: Convert the logarithmic equation to exponential form. If _b x = y, then b^y = x. Step 2: Substitute the given values _10 x = 0. x = 10^0 Step 3: Evaluate the exponential expression. Any non-zero number raised to the power of 0 is 1. x = 1 B) 1 11. Step 1: Recognize the expression as a difference of squares. The formula is (a+b)(a-b) = a^2 - b^2. Step 2: Substitute a=4 and b=sqrt(14). (4 + sqrt(14))(4 - sqrt(14)) = 4^2 - (sqrt(14))^2 = 16 - 14 = 2 A) 2 12. Step 1: Multiply the coefficients and the imaginary units. (-5i)(-2i) = (-5) × (-2) × i × i = 10i^2 Step 2: Substitute the value of i^2. Recall that i^2 = -1. = 10(-1) = -10 A) -10 13. The two coordinate axes (x-axis and y-axis) in a standard Cartesian system are perpendicular to each other, meaning they intersect at an angle of 90 degrees. B) 90^ 14. Step 1: Factorize both expressions. x^4 - y^4 = (x^2)^2 - (y^2)^2 = (x^2 - y^2)(x^2 + y^2) x^2 - y^2 Step 2: Identify the common and unique factors for the Least Common Multiple (LCM). The factors are (x^2 - y^2) and (x^2 + y^2). The LCM includes all factors raised to their highest power. LCM = (x^2 - y^2)(x^2 + y^2) Step 3: Simplify the LCM. LCM = x^4 - y^4 C) x^4 - y^4 15. A surd is an irrational number that can be expressed as a root of an integer. sqrt(5) is an irrational number because 5 is not a perfect square. Since it consists of a single term, it is a monomial surd. A) Monomial surd Got more? Send 'em!