Simplifying Algebraic Expressions: Step-by-Step Guide

To simplify the given algebraic expression, we will factorize the numerator and the terms in the denominator and then cancel out common factors.

The expression to simplify is:

\(\rm {256x^4 - 16y^4} \over {(80x^2 - 20y^2)}{(16x^2 + 4y^2)}\)

Step 1: Factor the Numerator

The numerator is \(256x^4 - 16y^4\). This is a difference of squares, \((a^2 - b^2) = (a-b)(a+b)\), where \(a^2 = 256x^4\) and \(b^2 = 16y^4\).

• \(a = \sqrt{256x^4} = 16x^2\)
• \(b = \sqrt{16y^4} = 4y^2\)

So, the numerator factors as:

\(256x^4 - 16y^4 = (16x^2 - 4y^2)(16x^2 + 4y^2)\)

Step 2: Factor the Denominator Terms

The denominator is \((80x^2 - 20y^2)(16x^2 + 4y^2)\).

Let's factor each term separately:

• First term: \(80x^2 - 20y^2\). We can factor out the greatest common divisor of 80 and 20, which is 20.

  \(80x^2 - 20y^2 = 20(4x^2 - y^2)\)

• Second term: \(16x^2 + 4y^2\). We can factor out the greatest common divisor of 16 and 4, which is 4.

  \(16x^2 + 4y^2 = 4(4x^2 + y^2)\)

So, the denominator becomes:

\([20(4x^2 - y^2)][4(4x^2 + y^2)]\)

However, looking back at the original expression, the second term in the denominator is given as \((16x^2 + 4y^2)\) not factored. Let's use the terms as given in the question for substitution first.

The denominator is \((80x^2 - 20y^2)(16x^2 + 4y^2)\).

Factor just the first term:

\(80x^2 - 20y^2 = 20(4x^2 - y^2)\)

Step 3: Substitute Factored Forms into the Expression

Substitute the factored numerator and the factored first term of the denominator back into the original expression:

\(\rm {(16x^2 - 4y^2)(16x^2 + 4y^2)} \over {[20(4x^2 - y^2)](16x^2 + 4y^2)}\)

Step 4: Cancel Common Factors

Observe that the term \((16x^2 + 4y^2)\) appears in both the numerator and the denominator. We can cancel this common factor.

The expression simplifies to:

\(\rm {16x^2 - 4y^2} \over {20(4x^2 - y^2)}\)

Step 5: Simplify the Remaining Expression

Now we simplify the remaining fraction:

Numerator: \(16x^2 - 4y^2\). We can factor out 4:

\(16x^2 - 4y^2 = 4(4x^2 - y^2)\)

Denominator: \(20(4x^2 - y^2)\)

Substitute this factored numerator back into the simplified expression:

\(\rm {4(4x^2 - y^2)} \over {20(4x^2 - y^2)}\)

Now, we can cancel the common factor \((4x^2 - y^2)\) from the numerator and the denominator.

This leaves us with a numerical fraction:

\(\rm {4} \over {20}\)

Step 6: Reduce the Numerical Fraction

Simplify the fraction \(\rm {4} \over {20}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 4.

\(\rm {4 \div 4} \over {20 \div 4} = {1 \over 5}\)

Thus, the simplified expression is \(\rm {1 \over 5}\).

Revision Table: Key Concepts in Algebraic Simplification

Additional Information: Importance of Factorization in Algebra

Factorization is a fundamental technique in algebra that helps in simplifying expressions, solving equations, and analyzing functions. When dealing with algebraic fractions, factorization is often the first step towards simplification. By expressing the numerator and denominator as products of their factors, we can easily identify and cancel out common terms, which significantly reduces the complexity of the expression.

Recognizing common factoring patterns like the difference of squares is crucial. Factoring out the greatest common divisor from terms is also essential for simplifying expressions effectively. These techniques allow us to transform complex algebraic fractions into simpler, equivalent forms, making further calculations or analysis much easier.