Understanding the Problem

We are asked to find the value of a given algebraic expression when the variable \(b\) has a specific value. The question specifically asks us to use an identity to solve this problem. The expression is:

\left(\frac{5}{b} + 5b\right)\left(\frac{25}{b^2} - 25 + 25b^2\right)

We are given that \(b = 5\).

Identifying the Correct Algebraic Identity

Let's look closely at the structure of the expression. It has two parts multiplied together. The first part is a sum of two terms, and the second part contains three terms. This structure resembles the expansion of the sum of cubes or the difference of cubes identity.

The sum of cubes identity is given by:

x^3 + y^3 = (x+y)(x^2 - xy + y^2)

Let's see if our expression fits this pattern. Let's consider the terms in the first parenthesis:

• Let x = \frac{5}{b}
• Let y = 5b

Now, let's square these terms and find their product to see if they match the terms in the second parenthesis:

• x^2 = \left(\frac{5}{b}\right)^2 = \frac{25}{b^2}
• y^2 = (5b)^2 = 25b^2
• xy = \left(\frac{5}{b}\right)(5b) = 5 \times 5 \times \frac{b}{b} = 25 \times 1 = 25

Now compare these results with the second parenthesis of the given expression: \left(\frac{25}{b^2} - 25 + 25b^2\right).

We see that the second parenthesis is exactly x^2 - xy + y^2, where x = \frac{5}{b} and y = 5b.

Therefore, the given expression is in the form (x+y)(x^2 - xy + y^2), which simplifies to x^3 + y^3.

Simplifying the Expression Using the Identity

Using the sum of cubes identity, the expression becomes:

\left(\frac{5}{b} + 5b\right)\left(\frac{25}{b^2} - 25 + 25b^2\right) = \left(\frac{5}{b}\right)^3 + (5b)^3

Substituting the Value of b

We are given that b=5. Now substitute this value into the simplified expression:

\left(\frac{5}{5}\right)^3 + (5 \times 5)^3

Calculating the Final Value

Now, we evaluate the expression with the substituted value of b:

• First term: \left(\frac{5}{5}\right)^3 = (1)^3 = 1
• Second term: (5 \times 5)^3 = (25)^3

To calculate 25^3:

25^3 = 25 \times 25 \times 25

25 \times 25 = 625

625 \times 25 = 15625

So, the expression simplifies to 1 + 15625.

1 + 15625 = 15626

Step-by-Step Evaluation

1. Identify the expression: \left(\frac{5}{b} + 5b\right)\left(\frac{25}{b^2} - 25 + 25b^2\right).
2. Recognize that the expression matches the form (x+y)(x^2 - xy + y^2) with x = \frac{5}{b} and y = 5b.
3. Apply the sum of cubes identity x^3 + y^3 = (x+y)(x^2 - xy + y^2) to simplify the expression to \left(\frac{5}{b}\right)^3 + (5b)^3.
4. Substitute the given value b=5 into the simplified expression: \left(\frac{5}{5}\right)^3 + (5 \times 5)^3.
5. Evaluate the terms: (1)^3 + (25)^3.
6. Calculate the cubes: 1^3 = 1 and 25^3 = 15625.
7. Add the results: 1 + 15625 = 15626.

The value of the expression when b=5 is 15626.

Revision Table: Key Concepts for Expression Evaluation

Additional Information on Algebraic Identities

Algebraic identities are powerful tools in mathematics that help simplify complex expressions and solve equations more easily. Recognizing these patterns is a key skill in algebra. Besides the sum of cubes, other important identities include:

• Difference of Squares: a^2 - b^2 = (a-b)(a+b)
• Perfect Square Trinomials: (a+b)^2 = a^2 + 2ab + b^2 and (a-b)^2 = a^2 - 2ab + b^2
• Difference of Cubes: a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Practicing identifying these patterns within various expressions will improve your ability to simplify and evaluate them efficiently.