Using the property of exponents that states a^m * a^n = a^(m+n), we combine the exponents: 2x + (x + 1) = 3x + 1.

log_10(0.01) can be rewritten as log_10(10^-2) = -2.

log_10(1/100) = log_10(10^-2) = -2.

log_10(100) = 2 and log_10(10) = 1, so 2 + 1 = 3.

Since 1000 is 10^3, log_10(1000) = 3.

Using the property of logarithms, log_5(25) = 2 and log_5(5) = 1. Therefore, log_5(25) + log_5(5) = 2 + 1 = 3, which is log_5(125).

Using the Binomial Theorem, the coefficient of x^3 is given by C(5,3) * (2^3) * (3^2) = 10 * 8 * 9 = 720.

Both 1/4^x and 4^(-x) represent the inverse of 4^x.

The product (x + 1)(x - 1) is a difference of squares, resulting in x^2 - 1.

The product of the roots of a quadratic polynomial ax^2 + bx + c is given by c/a. Here, c = 4 and a = 1, so the product is 4.

To find the sum of the coefficients, substitute x = 1 into the expression: (2(1) - 3)^4 = (-1)^4 = 1, but the coefficients sum to -81.

The sum of the roots is given by -b/a, which is -3/5 for this equation.

The vertex of the quadratic equation is given by the point (-b/2a, f(-b/2a)).

An exponential function approaches the x-axis as x approaches negative infinity but never actually touches it.

The function f(x) = 1/(x - 2) has a vertical asymptote at x = 2, where the function is undefined.

A horizontal line represents a function that is constant, meaning it neither increases nor decreases.

A parabola opening upwards is not one-to-one because it fails the horizontal line test; a horizontal line can intersect it at two points.

A strictly increasing function is one where, for any two points x1 and x2, if x1 < x2, then f(x1) < f(x2), which is represented by a graph that slopes upward from left to right.

A quadratic function is represented by a parabola, which can open either upwards or downwards.

A converging series approaches a finite limit as more terms are added.

A linear equation in two variables always forms a straight line when graphed.

Polynomial functions are continuous and smooth, meaning they do not have breaks, holes, or sharp corners.

A unique solution occurs when the equations intersect at exactly one point.

Composite numbers are defined as natural numbers greater than 1 that are not prime, meaning they have more than two distinct positive divisors.

Exponential functions grow or decay at a constant percentage rate, which is a defining characteristic.

Harmonic progression can include fractions, as it is defined through the reciprocals of an arithmetic progression.

Harmonic progressions can contain negative numbers, but the terms themselves must be positive to maintain the definition of harmonic sequences.

Irrational numbers are characterized by their non-repeating, non-terminating decimal expansion, distinguishing them from rational numbers.

A common rule for checking divisibility by 7 is to double the last digit and subtract it from the rest of the number; if the result is divisible by 7, then the original number is also divisible by 7.

The decimal system is a base-10 system, which is widely used in everyday calculations.