Algebraic Expressions and Identities

Find the product of $a^2$, $a^3$, and $a^4$.

Simplify the expression: $10 - (2 - x)$.

Find the sum of the expressions $(x + 1)$, $(x + 2)$, and $(x + 3)$.

Divide the monomial $50x^3y^2$ by $25x^2y^2$.

Simplify using identities: $(\frac{x}{2} + \frac{y}{3})(\frac{x}{2} - \frac{y}{3})$.

Subtract the product of $(x + 1)(x + 2)$ from the expression $x^2 + 4x + 5$.

Find the value of $p$ for which $x^2 - px + 16$ is a perfect square trinomial (where $p > 0$).

Simplify the product using identities: $(x + 1)(x^2 - x + 1)$.

If $ab = 6$ and $a + b = 5$, find the value of $a^2 + b^2$.

Simplify the expansion: $(x + 2)^3 - (x - 2)^3$.

Find the product $(2x + 5)(2x + 1)$.

Factorize $x^2 - 0.04$.

Simplify $(x+y)(x-y)(x^2+y^2)$.

Expand $(4 - 3a)^2$.

If $x - \frac{1}{x} = \sqrt{5}$, find the value of $x^2 + \frac{1}{x^2}$.

Evaluate $8.63^2 - 1.37^2$ using the identity $a^2 - b^2$.

Find the product $(x - \frac{1}{2})(x + \frac{1}{4})$ using the identity $(x+a)(x+b)$.

Evaluate $\frac{97 \times 97 - 3 \times 3}{94}$.

If $x^2 + \frac{1}{x^2} = 23$, find the value of $(x + \frac{1}{x})$.

Simplify $(3x + 4y)^2 - (3x - 4y)^2$.

Find the product of $(x + 0.4)$ and $(x + 0.5)$ using identity.

Factorize the expression $x^2 - 100$.

Expand $(3a - 4b)^2$.

Find the expansion of $(2x + 7)^2$.

If $x^2 + y^2 = 25$ and $x - y = 1$, find the value of $xy$.

Find the value of $997^2$ using the identity $(a-b)^2 = a^2 - 2ab + b^2$.

Expand $(\frac{3}{5}x + \frac{5}{3}y)^2$.

If $x + y = 10$ and $x - y = 4$, calculate the numerical value of the expression $x^3y - xy^3$.

The area of a square garden is given by the expression $(9x^2 + 24xy + 16y^2)$ square units. A new square garden is constructed such that its side length is 3 units longer than the side length of the original garden. Which of the following expressions represents the perimeter of the new garden?

If $a^2 + \frac{1}{a^2} = 102$, find the positive value of $a - \frac{1}{a}$.