Simplification of (7m-8n)^2:

To simplify the given expression, we can use the formula for (a-b)^2 which is a^2 - 2ab + b^2. Here, a = 7m and b = 8n.

So, substituting the values in the formula, we get:

(7m - 8n)^2 = (7m)^2 - 2(7m)(8n) + (8n)^2
= 49m^2 - 112mn + 64n^2

Therefore, the simplified form of (7m-8n)^2 is 49m^2 - 112mn + 64n^2.

Explanation:

The formula for (a-b)^2 is an algebraic identity that helps in simplifying expressions that involve squares of binomials. It is given by (a-b)^2 = a^2 - 2ab + b^2, where a and b are any real numbers.

In the given expression, (7m-8n)^2, we can use the formula by substituting a = 7m and b = 8n. This gives us:

(7m-8n)^2 = (7m)^2 - 2(7m)(8n) + (8n)^2

Simplifying the expression further, we get:

(7m)^2 = 49m^2
- 2(7m)(8n) = -112mn
(8n)^2 = 64n^2

Substituting these values in the above formula, we get:

(7m-8n)^2 = 49m^2 - 112mn + 64n^2

This is the simplified form of the given expression.

Algebraic Identities:

Algebraic identities are mathematical expressions that are true for any values of the variables involved. These identities are useful in simplifying complex algebraic expressions by substituting the values of the variables. Some of the commonly used algebraic identities are:

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

These identities are widely used in algebraic expressions to simplify them and solve equations involving variables.

=[(7m)²-2(7m)(8n)+(8n)²]+[(7m)²+ 2(7m)(8n)+(8n)²
=49m²-112mn+64n+49m²+112mn+64n²
=(49+49)m²+(64+64)n²(-112+112)mn
=98m²+128n²(0)
=98m²+128n²