Evaluate the given mathematical expression:

$\frac{{\left(0.8\right)}^4-{\left(0.2\right)}^4}{{\left(0.8\right)}^2+{\left(0.2\right)}^2}=?$

The correct answer is 0.60.

To evaluate the given mathematical expression efficiently, we can use algebraic factorization rather than expanding fourth powers directly.

Let us recall the algebraic identity for the difference of squares:

$a^2-b^2=(a+b)(a-b)$

We are given the following expression to evaluate:

$\frac{{\left(0.8\right)}^4-{\left(0.2\right)}^4}{{\left(0.8\right)}^2+{\left(0.2\right)}^2}$

Step 1: Rewrite the numerator using the difference of squares identity.

Notice that each term in the numerator can be expressed as a square of a squared value:

${\left(0.8\right)}^4={\left[{\left(0.8\right)}^2\right]}^2$

${\left(0.2\right)}^4={\left[{\left(0.2\right)}^2\right]}^2$

Let us define:

$x={\left(0.8\right)}^2$

$y={\left(0.2\right)}^2$

Applying the identity $x^2-y^2=(x+y)(x-y)$ to the numerator yields:

${\left(0.8\right)}^4-{\left(0.2\right)}^4=[{\left(0.8\right)}^2+{\left(0.2\right)}^2][{\left(0.8\right)}^2-{\left(0.2\right)}^2]$

Step 2: Substitute the factored numerator back into the expression.

$\frac{{\left(0.8\right)}^4-{\left(0.2\right)}^4}{{\left(0.8\right)}^2+{\left(0.2\right)}^2}=\frac{[{\left(0.8\right)}^2+{\left(0.2\right)}^2][{\left(0.8\right)}^2-{\left(0.2\right)}^2]}{{\left(0.8\right)}^2+{\left(0.2\right)}^2}$

Step 3: Simplify by canceling common factors.

The term ${\left(0.8\right)}^2+{\left(0.2\right)}^2$ is present in both the numerator and the denominator. Canceling it out leaves:

$={\left(0.8\right)}^2-{\left(0.2\right)}^2$

Step 4: Compute the simplified value.

We can find the final value using either of two methods:

Method 1 (Direct Squaring):

${\left(0.8\right)}^2=0.64$

${\left(0.2\right)}^2=0.04$

$0.64-0.04=0.60$

Method 2 (Using Difference of Squares again):

${\left(0.8\right)}^2-{\left(0.2\right)}^2$

$=(0.8+0.2)(0.8-0.2)$

$=\left(1.0\right)\left(0.6\right)=0.60$

Hence, the simplified value of the given expression is 0.60.