Question

Rationalize the numerator: \(\frac{\sqrt{x+7}-10}{x}\)

Short answer

The rationalized form is \(\frac{x - 93}{x (\sqrt{x+7} + 10)}\).

Step-by-step solution

Identify the conjugate

The conjugate of the numerator \(\sqrt{x+7} - 10\) is \(\sqrt{x+7} + 10\).

Multiply by the conjugate

Multiply both the numerator and the denominator by \(\sqrt{x+7} + 10\): \[\frac{(\sqrt{x+7} - 10) (\sqrt{x+7} + 10)}{x (\sqrt{x+7} + 10)}.\]

Simplify the numerator

Use the difference of squares formula, \(a^2 - b^2 = (a-b)(a+b)\), to simplify the numerator: \[(\sqrt{x+7})^2 - 10^2 = x+7 - 100 = x - 93.\]

Combine and simplify

Now the expression simplifies to: \[\frac{x - 93}{x (\sqrt{x+7} + 10)}.\]

Key concepts

conjugate

In mathematics, the term 'conjugate' has a special meaning, particularly when it comes to simplifying expressions involving square roots or other complex numbers. A conjugate involves changing the sign between two terms. For instance, the conjugate of \( a + b \) is \( a - b \). In our exercise, the conjugate of \( \sqrt{x+7} - 10 \) is \( \sqrt{x+7} + 10 \).

This technique is useful because multiplying conjugates results in a difference of squares. This process removes the square root, thereby simplifying the expression.

difference of squares

The 'difference of squares' is an essential algebraic identity. It is expressed as: \[ a^2 - b^2 = (a - b)(a + b). \] This formula states that the difference between two squares can be factored into the product of the sum and difference of the terms.

In our exercise, multiplying \( \sqrt{x+7} - 10 \) by \( \sqrt{x+7} + 10 \) uses this identity. Here's how it works:
The product \( (\sqrt{x+7} - 10)(\sqrt{x+7} + 10) \) simplifies to \[ (\sqrt{x+7})^2 - 10^2 = x + 7 - 100 = x - 93. \]
This transformation is crucial for simplifying complex rational expressions.

simplification

Simplification is the process of reducing an expression into a simpler or more manageable form. In the context of our exercise, the numerator is simplified using the difference of squares, and then we break down the whole fraction.

Initially, we had: \[ \frac{\sqrt{x+7} - 10}{x}. \]
By multiplying by the conjugate, the numerator becomes \( x - 93 \), resulting in: \[ \frac{x - 93}{x (\sqrt{x+7} + 10)}. \]
This form is simpler because it no longer includes a square root in the numerator.

rational expressions

A 'rational expression' is a fraction in which both the numerator and the denominator are polynomials. Simplifying rational expressions involves factoring and reducing them to their simplest form.

In our exercise, \ \frac{\sqrt{x+7} - 10}{x} \ is initially complicated because of the square root. By rationalizing the numerator and employing techniques like multiplication by the conjugate and the difference of squares, we made the expression more straightforward: \[ \frac{x - 93}{x (\sqrt{x+7} + 10)}. \]
This final form is a much clearer and more standard rational expression.