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Some common algebraic identities include:

1. (a + b)^2 = a^2 + 2ab + b^2 )

- Used to expand a square of a binomial.

2. (a - b)^2 = a^2 - 2ab + b^2 )

- Similar to the previous one but used when subtracting.

3. ( a^2 - b^2 = (a + b)(a - b)

- Called the difference of squares.

4. (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 )

- Used to expand a cube of a binomial.

5. (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 )

- Similar to the previous one but used when subtracting.

6. ( a^3 + b^3 = (a + b)(a^2 - ab + b^2)

- Sum of cubes.

These identities are used in various algebraic manipulations, simplifications, and factorizations. For example, when solving equations, simplifying expressions, or proving mathematical statements, these identities can be applied to make calculations more manageable.

Answer:

ALL IDENTITIES WHICH ARE USED TO SOLVE ALGEBRAIC EXPRESSION ARE AS FOLLOWS -:

1. Distributive Property:

• a(b+c) = ab + ac

This identity is used when we need to simplify expressions by distributing a factor to each term inside parentheses.

2. Commutative Property:

• a+b = b+a and ab = ba

This identity is used when we need to change the order of terms or factors in an expression without changing its value.

3. Associative Property:

• (a+b)+c = a+(b+c) and (ab)c = a(bc)

This identity is used when we need to regroup terms or factors in an expression without changing its value.

4. Identity Property:

• a+0 = a and a*1 = a

This identity is used when we need to add or multiply a term by a value that does not change its value.

5. Inverse Property:

• a+(-a) = 0 and a*(1/a) = 1

This identity is used when we need to find the additive or multiplicative inverse of a term.

6. Quadratic Formula:

• x = (-b ± √(b^2 - 4ac)) / 2a

This identity is used to solve quadratic equations of the form ax^2 + bx + c = 0.

7. Pythagorean Identity:

• sin^2θ + cos^2θ = 1

This identity is used in trigonometry to relate the sine and cosine of an angle in a right triangle.

8. Logarithmic Identity:

• loga(xy) = loga(x) + loga(y)

This identity is used in logarithmic equations to simplify expressions by breaking them down into smaller parts.

9. Exponential Identity:

• ab * ac = ab+c

This identity is used in exponential equations to simplify expressions by adding exponents with the same base.

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