Question

Find a conjugate of each expression and the product of the expression with the conjugate. $$ b \sqrt{a}+a \sqrt{b} $$

Short answer

The conjugate of the given expression is \(b \sqrt{a} - a \sqrt{b}\), and the product of the expression with its conjugate is \(b^2 a - a^2 b\).

Step-by-step solution

Finding the conjugate

Find the conjugate of the given expression by changing the sign of the imaginary part (in this case, the square roots). The conjugate of \(b \sqrt{a} + a \sqrt{b}\) is: $$ b \sqrt{a} - a \sqrt{b}$$

Calculating the product of the expression and its conjugate

Multiply the given expression by its conjugate. $$(b \sqrt{a} + a \sqrt{b})(b \sqrt{a} - a \sqrt{b})$$

Applying the difference of squares formula

Apply the difference of squares formula for the product: $$(A+B)(A-B) = A^2 - B^2$$where \(A = b \sqrt{a}\) and \(B = a \sqrt{b}\). The product becomes: $$ (b \sqrt{a})^2 - (a \sqrt{b})^2$$

Simplifying the expressions

Simplify the squared terms in the product: $$(b^2 a) - (a^2 b)$$ The conjugate of the given expression is \(b \sqrt{a} - a \sqrt{b}\), and the product of the expression with its conjugate is \(b^2 a - a^2 b\).

Key concepts

Conjugate of Complex Expressions

In mathematics, the term "conjugate" is often associated with complex numbers or expressions involving square roots and can play a crucial role in simplification.
The conjugate of an expression involves changing the sign of certain terms. In the context of complex numbers, given a complex number like \(z = a + bi\), the conjugate \(\overline{z}\) is \(a - bi\). This dual formation negates the imaginary component.
For expressions involving radicals, like our exercise \(b \sqrt{a} + a \sqrt{b}\), it follows a similar principle. Here, since both parts involve square roots, the imaginary "nature" is handled similarly by altering the sign between terms:

• The expression \(b \sqrt{a} + a \sqrt{b}\) turns into \(b \sqrt{a} - a \sqrt{b}\), by flipping the sign.

This alteration is critical, especially when working toward rationalizing the expression or simplifying forms that involve these non-polynomial terms.

Difference of Squares Formula

The difference of squares formula is a mathematical tool that helps simplify a particular form of polynomial multiplication.
The general formula is \((A+B)(A-B) = A^2 - B^2\). This effectively removes the middle terms when multiplying two binomials.
Applying this in the exercise involves understanding what \(A\) and \(B\) represent in our given expression. Here, identify \(A = b \sqrt{a}\) and \(B = a \sqrt{b}\). When you multiply \((b \sqrt{a} + a \sqrt{b})(b \sqrt{a} - a \sqrt{b})\), using difference of squares results in:

• \( (b \sqrt{a})^2 - (a \sqrt{b})^2 \)

This expression simplifies the multiplication process by only focusing on the squares of the terms \(A\) and \(B\), omitting any cross-product terms which would cancel out.

Expressions with Radicals

Expressions with radicals can initially seem complex, but understanding how to manage and simplify them is key.
Radicals, like square roots, introduce an "irrational" component. This characteristic can make direct computation challenging. When encountered within expressions, simplifying them often involves rationalizing.
In the context of our problem, radicals such as \(b \sqrt{a}\) and \(a \sqrt{b}\) each contain products that are essentially part of a larger binomial expression.

• When squared, each calculates the product within the square root. For example, \((b \sqrt{a})^2\) becomes \(b^2 a\) because \((b \sqrt{a})^2 = b^2 \cdot a\).

Understanding this characteristic helps in simplifying expressions, as seen in our final step: simplifying \((b \sqrt{a})^2 - (a \sqrt{b})^2\) to \(b^2 a - a^2 b\). Using these simplification processes helps clear the radical expression into something more manageable.