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Chapter 10: Problem 30

Simplify using the quotient rule. $$\sqrt{\frac{19}{25}}$$

Short Answer

Expert verified
So, the simplified form of the expression \(\sqrt{\frac{19}{25}}\) using the quotient rule is \(\frac{\sqrt{19}}{5}\).

Step by step solution

01

Apply quotient rule

First, apply the quotient rule, which states that the square root of a fraction is equivalent to the square root of the numerator divided by the square root of the denominator, to the given problem. So, let us rewrite the given expression \(\sqrt{\frac{19}{25}}\) as \(\frac{\sqrt{19}}{\sqrt{25}}\).
02

Simplify the square root terms

Now, let us simplify the square root terms. Square root of 25 is a real number 5, because \(5^2 = 25\). As for \(\sqrt{19}\), since 19 is a prime number and does not have a perfect square root, it stays as \(\sqrt{19}\). So the simplified expression is \(\frac{\sqrt{19}}{5}\)

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Key Concepts

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

Quotient rule for radicals
The *quotient rule for radicals* is a fundamental concept in simplifying expressions that involve square roots and fractions. This rule makes working with complex expressions more straightforward.
When you encounter a radical surrounding a fraction, the rule states:
  • The square root of a fraction is the quotient of the square roots of the numerator and the denominator.
  • Mathematically, this is represented as \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\), provided \(b eq 0\).
The quotient rule helps in simplifying expressions, making them easier to handle, especially in calculations. For example, in the expression \(\sqrt{\frac{19}{25}}\), we apply the rule to break it down into \(\frac{\sqrt{19}}{\sqrt{25}}\), which allows further simplification.
Prime numbers
Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. They are the building blocks of number theory, as every integer greater than 1 is either a prime or can be factored into primes.
  • A few examples of prime numbers are 2, 3, 5, 7, 11, 13, 17, and 19.
  • Prime numbers help in simplifying square roots when checking if a square root can be evaluated to a whole number.
In our original problem, \(19\) is identified as a prime number, which means it cannot be broken down into any smaller factors other than 1 and 19. Therefore, since 19 is not a perfect square, \(\sqrt{19}\) remains in its simplest radical form.
Square roots
A square root of a number is a value that, when multiplied by itself, gives the original number. Working with square roots often involves simplifying them whenever possible.
The square root symbol \(\sqrt{}\) signifies this operation.
  • A perfect square is a number whose square root is an integer, for example, \(\sqrt{25} = 5\) because \(5 \times 5 = 25\).
  • If a number is not a perfect square, like \(\sqrt{19}\), the radical remains as it is unless it can be simplified further using properties of radicals.
Square roots are pivotal when dealing with radical expressions, as seen in the expression \(\sqrt{\frac{19}{25}}\), where applying the quotient rule and simplifying \(\sqrt{25}\) to 5 allows us to simplify the entire expression.

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Most popular questions from this chapter

The early Greeks believed that the most pleasing of all rectangles were golden rectangles, whose ratio of width to height is \(\frac{w}{h}=\frac{2}{\sqrt{5}-1}\) The Parthenon at Athens fits into a golden rectangle once the triangular pediment is reconstructed. (PICTURE NOT COPY) Rationalize the denominator of the golden ratio. Then use a calculator and find the ratio of width to height, correct to the nearest hundredth, in golden rectangles.

In Exercises \(65-74,\) simplify each radical expression and then rationalize the denominator. $$\frac{9}{\sqrt{3 x^{2} y}}$$

In Exercises \(65-74,\) simplify each radical expression and then rationalize the denominator. $$\frac{12}{\sqrt[3]{-8 x^{5} y^{8}}}$$

Describe how to multiply conjugates.

In Exercises \(65-74,\) simplify each radical expression and then rationalize the denominator. $$\sqrt{\frac{7 m^{2} n^{3}}{14 m^{3} n^{2}}}$$
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