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Chapter 12: Problem 78

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

Short Answer

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

Step by step solution

01

Identify the conjugate

The conjugate of the numerator \(\sqrt{x+7} - 10\) is \(\sqrt{x+7} + 10\).
02

Multiply by the conjugate

Multiply both the numerator and the denominator by \(\sqrt{x+7} + 10\): \[\frac{(\sqrt{x+7} - 10) (\sqrt{x+7} + 10)}{x (\sqrt{x+7} + 10)}.\]
03

Simplify the numerator

Use the difference of squares formula, \(a^2 - b^2 = (a-b)(a+b)\), to simplify the numerator: \[(\sqrt{x+7})^2 - 10^2 = x+7 - 100 = x - 93.\]
04

Combine and simplify

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Mixing Nuts A store sells cashews for \(\$ 5.00\) per pound and peanuts for \(\$ 1.50\) per pound. The manager decides to mix 30 pounds of peanuts with some cashews and sell the mixture for \(\$ 3.00\) per pound. How many pounds of cashews should be mixed with the peanuts so that the mixture will produce the same revenue as selling the nuts separately?

Carletta has \(\$ 10,000\) to invest. As her financial consultant, you recommend that she invest in Treasury bills that yield \(6 \%,\) Treasury bonds that yield \(7 \%\) and corporate bonds that yield \(8 \% .\) Carletta wants to have an annual income of \(\$ 680\), and the amount invested in corporate bonds must be half that invested in Treasury bills. Find the amount in each investment.

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{rr} 2 x+y-3 z= & 0 \\ -2 x+2 y+z= & -7 \\ 3 x-4 y-3 z= & 7 \end{array}\right. $$

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Simplify: \(\left(\frac{18 x^{4} y^{5}}{27 x^{3} y^{9}}\right)^{3}\)

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{aligned} 2 x-y &=-1 \\ x+\frac{1}{2} y &=\frac{3}{2} \end{aligned}\right. $$
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