Question

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{x+h}-\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}$$

Short answer

The rationalized form is \(\frac{2x + h - 2\sqrt{x(x+h)}}{h}\).

Step-by-step solution

Recognize the Need to Rationalize

The given expression \(\frac{\sqrt{x+h}-\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}\) has a denominator with a sum of square roots. Rationalizing the denominator involves eliminating the square roots from the denominator.

Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{x+h}-\sqrt{x}\). This is the technique used to eliminate the radicals in the denominator.

Apply the Conjugate

Rewrite the expression: \(\frac{\sqrt{x+h}-\sqrt{x}}{\sqrt{x+h}+\sqrt{x}} \times \frac{\sqrt{x+h}-\sqrt{x}}{\sqrt{x+h}-\sqrt{x}} = \frac{(\sqrt{x+h}-\sqrt{x})^2}{(\sqrt{x+h})^2-(\sqrt{x})^2}\)

Simplify the Numerator

Simplify the numerator using the binomial square formula: \((a-b)^2 = a^2 - 2ab + b^2\). Here, \(a = \sqrt{x+h}\) and \(b = \sqrt{x}\): \((\sqrt{x+h}-\sqrt{x})^2 = x+h - 2\sqrt{x+h}\sqrt{x} + x\)

Simplify the Denominator

Simplify the denominator using the difference of squares formula: \(a^2 - b^2\). Here, \(a = \sqrt{x+h}\) and \(b = \sqrt{x}\): \((\sqrt{x+h})^2 - (\sqrt{x})^2 = (x+h)-x = h\)

Combine and Simplify

Combine the simplified numerator and denominator: \(\frac{(x+h)-2\sqrt{x(x+h)}+x}{h} = \frac{2x+h - 2\sqrt{x(x+h)}}{h}\)

Key concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. They can range from simple to complex. For example, in the expression \( \frac{(\frac{\beta _1}{-\beta _1})} \bum and #^\frac{H_b2^a} \), the parts with variables and numbers form an algebraic expression.
Understanding algebraic expressions is essential as they form the basis of many mathematical operations.
When you practice rationalizing denominators, you deal with algebraic expressions extensively.
Recognizing patterns and applying algebraic rules makes it easier to break down and solve these expressions.

Radicals

Radicals involve the root of a number, the most common being the square root.
The square root symbol \( \sqrt{} \) represents a number that, when multiplied by itself, yields the original number. For instance, \( \sqrt{25} = 5 \).
When dealing with algebraic expressions containing radicals, it's crucial to simplify them whenever possible.
For example, in the given exercise \( \frac{\beta _2}{4} \sqrt{x+h} and 1 \frac{var1} \), simplifying radicals helps in further calculations.

Conjugates

Conjugates are useful for rationalizing denominators.
If you have a sum or difference involving square roots in the denominator, you can multiply by its conjugate.
For example, the conjugate of \( (\frac{\beta _1}{r})/c)) \sqrt{x+h} + \sqrt{x} \) is \( \frac{\beta _1}{r(dicite, 1)} \sqrt{x+h} - \sqrt{x} \).
This technique helps eliminate the radicals in the denominator, making the expression easier to work with.
Understanding conjugates and how to apply them is essential for rationalizing denominators effectively.

Binomial Expansion

The binomial expansion formula helps to simplify expressions involving the sum or difference of two terms raised to a power.
The binomial expansion of \( a + b \) squared gives \( (0)2 - 2a*b + b^2 \).
In rationalizing denominators, we use this rule to simplify the numerator or denominator.
For instance, in our exercise, \( (\sqrt{x+h}-sqrt{x})^2 = x + h - 2*\sqrt{x}\sqrt{x} + x \). This simplification step allows for further reduction and easier manipulation of the expression.
Mastering this concept greatly aids in handling complex algebraic and radical expressions.