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Chapter 0: Problem 33

Simplify each expression involving square roots. $$ \frac{1}{\sqrt{3}} $$

Short Answer

Expert verified
\( \frac{\sqrt{3}}{3} \)

Step by step solution

01

- Rationalize the Denominator

To simplify the expression \( \frac{1}{\sqrt{3}} \), it is important to rationalize the denominator. This means removing the square root from the denominator by multiplying both the numerator and the denominator by the same square root. In this case, multiply \( \frac{1}{\sqrt{3}} \) by \( \frac{\sqrt{3}}{\sqrt{3}} \).
02

- Perform the Multiplication

Multiply the numerators and the denominators together: \[ \frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{3} \]
03

- Simplify the Expression

The denominator becomes a whole number, as \( \sqrt{3} \cdot \sqrt{3} = 3 \). Therefore, the expression simplifies to: \[ \frac{\sqrt{3}}{3} \]

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Key Concepts

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

Rationalizing the Denominator
When you have a fraction with a square root in the denominator, it's often useful to get rid of that square root. This process is called rationalizing the denominator.
Doing this makes the expression simpler and easier to work with.

Here鈥檚 how you rationalize the denominator:
  • Identify the square root in the denominator.
  • Multiply both the numerator and the denominator by that square root.
For example, take the fraction \(\frac{1}{\sqrt{3}}\). You multiply it by \(\frac{\sqrt{3}}{\sqrt{3}}\), which gives you \(\frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}\). This process ensures that the square root is removed from the denominator, simplifying the fraction.
Multiplication of Square Roots
When multiplying square roots, use the property that states \[ \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \]. This property makes it straightforward to handle square roots.
So, in our example, \(\frac{1}{\sqrt{3}}\), after multiplying by \(\frac{\sqrt{3}}{\sqrt{3}}\), this becomes \(\frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{3}\).

Notice how the multiplicative property of square roots turns \( \sqrt{3} \cdot \sqrt{3} \) in the denominator into 3. This simplification makes calculations easier and more practical.
Simplifying Fractions
Once you have the fraction with no square root in the denominator, you can simplify it if possible. In our example, after rationalizing the denominator, the expression became \( \frac{\sqrt{3}}{3} \).
Now, check if there are any common factors that can be divided out.
Here, \( \frac{\sqrt{3}}{3} \) is already in its simplest form, meaning no further simplification is needed.

Remember: Simplifying fractions always involves:
  • Finding the greatest common divisor (GCD) of the numerator and the denominator
  • Dividing both the numerator and the denominator by that GCD.
This ensures the fraction is as simple as possible. Keeping fractions simplified helps reduce errors and makes further calculations easier.

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