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Chapter 11: Problem 60

Write each expression in simplest radical form. If \(a\) radical appears in the denominator, rationalize the denominator. $$\sqrt{x^{2}+8^{-1}}$$

Short Answer

Expert verified
The simplified form is \(\frac{\sqrt{64x^2 + 8}}{8}\).

Step by step solution

01

Simplify Inside the Square Root

First, we will simplify the expression inside the square root. The given expression is \(\sqrt{x^2 + 8^{-1}}\). Recall that \(8^{-1}\) can be rewritten as \(\frac{1}{8}\). Therefore, the expression inside the square root becomes \(x^2 + \frac{1}{8}\).
02

Write as a Single Fraction

Next, we need to express \(x^2 + \frac{1}{8}\) as a single fraction. To do this, express \(x^2\) with a denominator of 8: \(\frac{8x^2}{8}\). This allows us to add these two terms: \(\frac{8x^2}{8} + \frac{1}{8} = \frac{8x^2 + 1}{8}\).
03

Simplify the Square Root Expression

Now, we take the square root of \(\frac{8x^2 + 1}{8}\). This can be expressed as a fraction: \(\sqrt{\frac{8x^2 + 1}{8}} = \frac{\sqrt{8x^2 + 1}}{\sqrt{8}}\).
04

Rationalize the Denominator

To remove the square root from the denominator, multiply the numerator and the denominator by \(\sqrt{8}\). Thus, \(\frac{\sqrt{8x^2 + 1}}{\sqrt{8}} \times \frac{\sqrt{8}}{\sqrt{8}} = \frac{\sqrt{8(8x^2 + 1)}}{8}\).
05

Finalize the Expression

Finally, simplify the numerator as much as possible. The expression becomes \(\frac{\sqrt{64x^2 + 8}}{8}\). Since 64 is a perfect square, this can further be expressed in simplest form as \(\frac{\sqrt{64x^2 + 8}}{8}\), but \(8\) cannot be simplified further inside the radical involving x.

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Key Concepts

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

Rationalizing the Denominator
Rationalizing the denominator is a crucial process in simplifying expressions containing square roots in the denominator. This method makes the expression easier to work with and avoids having irrational numbers in denominators.
To rationalize a denominator, we multiply both the numerator and the denominator by a value that will eliminate the square root from the denominator.
  • In our example, we had \(\frac{\sqrt{8x^2 + 1}}{\sqrt{8}}\) in the denominator.
  • By multiplying by \(\sqrt{8}\), the expression became: \(\frac{\sqrt{8x^2 + 1} \times \sqrt{8}}{\sqrt{8} \times \sqrt{8}} = \frac{\sqrt{64x^2 + 8}}{8}\).
After multiplying, the square root in the denominator is eliminated, and thus, rationalizing simplifies the expression by allowing us to handle fractions more effectively without improper roots.
Square Roots
Understanding square roots is key in simplifying radical expressions. A square root simply involves finding a number which, when multiplied by itself, gives the original number.
Square roots are common in algebra and are essential when dealing with expressions under a radical symbol.
  • The square root of a perfect square results in a whole number. For instance, \(\sqrt{64} = 8\) because 8 times 8 equals 64.
  • Numbers which aren't perfect squares have an irrational square root, like \(\sqrt{8}\) which can't be simplified into a non-decimal number.
Understanding how square roots work can help simplify expressions and make complex algebra tasks more manageable.
Fraction Operations
Fraction operations play an important role in working with expressions that involve denominators. They require a clear understanding of how to manage numerators and denominators.In the given exercise, transforming the expression inside the square root into a single fraction was essential:
  • We transformed \(x^2 + \frac{1}{8}\) into \(\frac{8x^2 + 1}{8}\) by converting \(x^2\) to \(\frac{8x^2}{8}\).
This process allowed us to work with a single fraction, simplifying further steps in handling the radical expression.
Being proficient with fraction operations ensures that more complex expressions can be managed without making errors that could lead to incorrect simplifications.

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Most popular questions from this chapter

Perform the indicated operations, expressing answers in simplest form with rationalized denominators. $$(3+\sqrt{6-7 a})(2-\sqrt{6-7 a})$$

Perform the indicated operations. Simplify \(\left(x^{n-1} \div x^{n-3}\right)^{1 / 3}\) and express the result as a radical.

Express each of the given expressions in simplest form with only positive exponents. $$\frac{a x^{-2}+a^{-2} x}{a^{-1}+x^{-1}}$$

Perform the indicated operations, expressing answers in simplest form with rationalized denominators. Then verify the result with a calculator. $$(\sqrt{11}+\sqrt{6})(\sqrt{11}-2 \sqrt{6})$$

$$\text {Solve the given problems.}$$ When analyzing the ratio of resultant forces when forces \(\mathbf{F}\) and \(\mathbf{T}\) act on a structure, the expression \(\sqrt{F^{2}+T^{2}} / \sqrt{F^{-2}+T^{-2}}\) arises. Simplify this expression.
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