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Chapter 9: Problem 1

Simplify the square root expressions. $$ \frac{\sqrt{26}}{\sqrt{13}} $$

Short Answer

Expert verified
Question: Simplify the expression \(\frac{\sqrt{26}}{\sqrt{13}}\). Answer: \(\sqrt{2}\)

Step by step solution

01

Apply the property of square roots

Using the property mentioned in the analysis, we have: $$ \frac{\sqrt{26}}{\sqrt{13}} = \sqrt{\frac{26}{13}} $$.
02

Simplify the fraction

Now we simplify the fraction inside the square root: $$ \sqrt{\frac{26}{13}} = \sqrt{\frac{2 \cdot 13}{13}} $$. Since 13 is a factor in both the numerator and denominator, we can cancel it: $$ \sqrt{\frac{2 \cdot 13}{13}} = \sqrt{2} $$.
03

Final answer

The simplified answer for the given expression is: $$ \frac{\sqrt{26}}{\sqrt{13}} = \sqrt{2} $$.

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Key Concepts

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

Properties of Square Roots
Square roots can often appear confusing, but they operate under distinct laws that make them more manageable. One such important property is the ability to simplify a square root that is divided by another square root into a single square root of the fraction \( \frac{a}{b} \). This means that if you encounter something like \( \frac{\sqrt{a}}{\sqrt{b}} \), you can convert it into \( \sqrt{\frac{a}{b}} \).
  • This property helps us to simplify complex expressions and make equations cleaner.
  • By dealing with the expression as a whole, you often minimize errors that could occur if managed piece by piece.
  • Letting us see the big picture, which can sometimes make other simplifications obvious.
Recognizing and using this property effectively can greatly help in simplifying mathematical expressions and solving algebra problems. It's about making the components work together to ease the calculations.
Fraction Simplification
Fraction simplification is a crucial concept when dealing with any kind of mathematical expressions, especially those involving square roots. In the exercise, the fraction \( \frac{26}{13} \) inside the square root was simplified by dividing both the numerator and the denominator by the common factor, 13.
  • The goal is to reduce fractions to their simplest form.
  • This involves identifying common factors between the numerator and denominator that can cancel each other out.
  • Here, 26 and 13 were both divisible by 13, leading to the simplified form of \( \frac{2 \cdot 13}{13} = 2 \).
Simplifying fractions is not only crucial in basic algebra but also forms the foundation for solving more complex algebraic problems involving radicals and other higher-level mathematics. It's a simple yet powerful tool to have in your problem-solving toolbox.
Radical Expressions
Radical expressions involve roots, with square roots being the most common. Simplifying these expressions can significantly streamline problem-solving processes. In this exercise, the expression began as a ratio of two square roots. By leveraging properties of radicals, it was transformed to a simpler form: \( \sqrt{2} \).
  • Radicals turn multiplication or division of numbers into roots, potentially simplifying calculations.
  • Recognizing when to combine roots under a single radical can help achieve simplification quickly and accurately.
  • Always aim to express radicals in their simplest form for clarity and ease in further mathematical processes.
Understanding radical expressions and how to manage them is a foundational skill required as mathematical complexity increases. When you encounter radicals in equations, look for opportunities to simplify them using properties of numbers and radicals for efficient solutions.

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Most popular questions from this chapter

Simplify each expression by performing the indicated operation. $$ (1+\sqrt{3})(2-\sqrt{3}) $$

Simplify each expression by performing the indicated operation. $$ (5+\sqrt{6})(4-\sqrt{6}) $$

For the following problems, simplify the expressions. $$ \sqrt{(8 a-5 b)^{26}(2 a-9 b)^{40}(a-b)^{15}} $$

Simplify each expression by performing the indicated operation. $$ \sqrt{2}(\sqrt{3}+1) $$

Simplify each expression by performing the indicated operation. $$ (1+\sqrt{3 x})^{2} $$
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