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Chapter 13: Problem 30

Simplify each expression. \(\frac{\sqrt{21}}{\sqrt{7}}\)

Short Answer

Expert verified
\(\sqrt{3}\)

Step by step solution

01

Identify the Given Expression

We need to simplify the expression \(\frac{\sqrt{21}}{\sqrt{7}}\). This expression is a fraction where both the numerator and the denominator are square roots.
02

Apply the Quotient Rule for Square Roots

The quotient rule for square roots states that \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\) when \(b eq 0\). Let's apply this rule to the expression: \(\sqrt{\frac{21}{7}}\).
03

Simplify the Fraction Inside the Square Root

Divide 21 by 7, which equals 3. So, \(\sqrt{\frac{21}{7}}\) simplifies to \(\sqrt{3}\).
04

Write the Simplified Expression

The simplified form of the original expression \(\frac{\sqrt{21}}{\sqrt{7}}\) is \(\sqrt{3}\).

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Key Concepts

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

Quotient Rule for Square Roots
When you encounter an expression where both the numerator and the denominator are square roots, the quotient rule for square roots can be very useful. Consider the expression \( \frac{\sqrt{21}}{\sqrt{7}} \). This expression can initially look complex, but the quotient rule simplifies this easily. The rule states:
  • For any positive real numbers \(a\) and \(b\), \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), provided \(b eq 0\).
This means we can merge the two square roots into one by dividing the values under the square roots. For our example, \(\frac{\sqrt{21}}{\sqrt{7}}\) becomes \(\sqrt{\frac{21}{7}}\), offering a simple way to handle expressions like these.
Fraction Simplification
Once you have applied the quotient rule for square roots, the next step is often to simplify the fraction under the square root. Simplification of fractions is about reducing them to their simplest form. To do this effectively, identify the greatest common divisor (GCD) of the numerator and the denominator and divide both by this number.
  • For our expression \(\sqrt{\frac{21}{7}}\), the fraction \(\frac{21}{7}\) simplifies to \(3\) because dividing 21 by 7 leaves us with 3.
Simplifying fractions is crucial because it reduces the numbers you are working with, making the calculations easier and the results more straightforward.
Square Root Properties
Understanding square root properties helps in simplifying expressions further. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).
  • One fundamental property is \(\sqrt{a^2} = a\), where \(a\) is positive.
  • Another helpful property is if two numbers are multiplied under a square root, they can be split: \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\).
For our final simplified example, \(\sqrt{3}\), we directly recognize \(3\) as a prime number, suggesting \(\sqrt{3}\) cannot be simplified further. Understanding these properties allows such insights and simplifications.

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