Evaluating Algebraic Expressions

This problem asks us to find the numerical value of a given algebraic expression when specific values are assigned to the variables x and y. Evaluating an expression means substituting the given values for the variables and then performing the indicated mathematical operations according to the order of operations (PEMDAS/BODMAS).

Step-by-Step Evaluation Process

The given expression is: \(16x^4 + 25y^2 - 40x^2y\)

The given values for the variables are:

• x = 5
• y = 2

Calculating Powers of Variables

First, let's calculate the required powers of x and y:

• Calculate \(x^2\): \(x^2 = 5^2 = 5 \times 5 = 25\)
• Calculate \(x^4\): \(x^4 = (x^2)^2 = (25)^2 = 25 \times 25 = 625\)
• Calculate \(y^2\): \(y^2 = 2^2 = 2 \times 2 = 4\)

Calculating Terms in the Expression

Now, let's calculate the value of each term in the expression using the calculated powers and the given coefficients:

• First term: \(16x^4 = 16 \times 625\)
• \(16 \times 625 = 10000\)
• Second term: \(25y^2 = 25 \times 4\)
• \(25 \times 4 = 100\)
• Third term: \(-40x^2y\)
• We need \(x^2y\). We know \(x^2 = 25\) and \(y = 2\).
• So, \(x^2y = 25 \times 2 = 50\)
• Now, calculate the third term: \(-40 \times 50\)
• \(-40 \times 50 = -2000\)

Substituting and Combining Terms

Substitute the calculated values of the terms back into the original expression:

\(16x^4 + 25y^2 - 40x^2y = (16x^4) + (25y^2) + (-40x^2y)\)

Substitute the calculated values:

\(10000 + 100 + (-2000)\)

\(10000 + 100 - 2000\)

Now, perform the addition and subtraction from left to right:

\(10000 + 100 = 10100\)

\(10100 - 2000 = 8100\)

Final Value of the Expression

The value of the expression \(16x^4 + 25y^2 - 40x^2y\) at x = 5 and y = 2 is 8100.

Comparing with Options

Let's compare our calculated value with the given options:

Our calculated value, 8100, matches Option 1.

Revision Table: Key Concepts in Expression Evaluation

Additional Information: Recognizing Patterns

Sometimes, algebraic expressions can be simplified or recognized as specific patterns (like perfect squares) before substituting values. Let's look at the given expression again:

\(16x^4 + 25y^2 - 40x^2y\)

Notice that \(16x^4 = (4x^2)^2\) and \(25y^2 = (5y)^2\).

The middle term is \(-40x^2y\). Let's see if it fits the form \(-2ab\) where \(a = 4x^2\) and \(b = 5y\).

\(-2ab = -2 \times (4x^2) \times (5y)\)

\(-2 \times 4x^2 \times 5y = -8x^2 \times 5y = -40x^2y\)

Yes, the middle term fits the pattern \(-2ab\).

So, the expression \(16x^4 + 25y^2 - 40x^2y\) is a perfect square trinomial of the form \(a^2 - 2ab + b^2\), which can be factored as \((a-b)^2\).

Here, \(a = 4x^2\) and \(b = 5y\).

The expression can be written as \((4x^2 - 5y)^2\).

Now, let's evaluate this factored form with x = 5 and y = 2:

First, calculate the value inside the parenthesis:

\(4x^2 - 5y\)

Substitute x = 5 and y = 2:

\(4(5^2) - 5(2)\)

\(4(25) - 10\)

\(100 - 10 = 90\)

Now, square the result:

\((90)^2 = 90 \times 90 = 8100\)

This confirms our previous calculation and shows that recognizing algebraic patterns can sometimes simplify the evaluation process.