Exploring the Value of (a^2 b^2) Given (a frac{3sqrt{5}}{2})

In mathematics, we often encounter problems that require us to find the value of a given expression based on the values of specific variables. This article aims to delve into the problem of finding the value of (a^2 b^2) when it is provided that (a frac{3sqrt{5}}{2}). While the value of (b) is not directly specified, we will explore the relationship between (a^2), (b^2), and the given value of (a). This exploration will be helpful in understanding algebraic expressions and mathematical equations.

Given Information: (a frac{3sqrt{5}}{2})

The problem begins with the information that (a frac{3sqrt{5}}{2}). This value of (a) is an example of a real number that involves a square root, which is a common occurrence in various mathematical and scientific contexts. The square root of 5 (which is approximately 2.236) combined with the ratio of 3 to 2 provides a specific value for (a).

Calculating (a^2)

To find (a^2), we simply need to square the given value of (a). Let's calculate it step-by-step:

[a frac{3sqrt{5}}{2}]

[a^2 left(frac{3sqrt{5}}{2}right)^2]

[a^2 frac{(3sqrt{5})^2}{2^2}]

[a^2 frac{9 cdot 5}{4}]

[a^2 frac{45}{4}]

[a^2 11.25]

Thus, the value of (a^2) is 11.25.

Understanding (a^2 b^2)

Given the value of (a^2), we can now address the problem of finding the value of (a^2 b^2). However, the value of (b) is not specified in the problem. This means that the expression (a^2 b^2) will have an infinite number of solutions, depending on the value of (b). To illustrate this, let's consider a few scenarios:

Scenario 1: (b 0)

If (b 0), then:

[a^2 b^2 11.25 0]

[a^2 b^2 11.25]

Thus, the value of (a^2 b^2) is 11.25 when (b 0).

Scenario 2: (b frac{5sqrt{2}}{2})

If (b frac{5sqrt{2}}{2}), then:

[a^2 b^2 11.25 left(frac{5sqrt{2}}{2}right)^2]

[a^2 b^2 11.25 frac{25 cdot 2}{4}]

[a^2 b^2 11.25 12.5]

[a^2 b^2 23.75]

Thus, the value of (a^2 b^2) is 23.75 when (b frac{5sqrt{2}}{2}).

General Case

In the general case, let's consider that (b) can take any real value. If (b) is any real number, then the value of (a^2 b^2) can be expressed as:

[a^2 b^2 11.25 b^2]

This expression shows that (a^2 b^2) varies directly with the square of (b). As (b) increases, the value of (a^2 b^2) increases, and vice versa. Similarly, if (b) is negative, the value of (b^2) will remain positive, and the overall value of (a^2 b^2) will still increase.

Conclusion

Given the problem of finding the value of (a^2 b^2) when (a frac{3sqrt{5}}{2}), we have demonstrated that the expression can take on an infinite number of values, depending on the value of (b). When (b 0), the value is 11.25, and for (b frac{5sqrt{2}}{2}), the value is 23.75. Recognizing this, we can conclude that the expression (a^2 b^2) can be useful in various mathematical and scientific contexts, including optimization problems and geometric interpretations.

Related Keywords

(Keyword 1: a squared plus b squared
Keyword 2: algebraic expressions
Keyword 3: mathematical equations)