Question

Simplify each expression. \(\frac{1}{\sqrt{8}}\)

Short answer

\(\frac{\sqrt{2}}{4}\)

Step-by-step solution

Simplify the Radical

We start by simplifying the square root in the denominator. We know that \(\sqrt{8} = \sqrt{4 \times 2}\). Since \(\sqrt{4} = 2\), we can rewrite it as \(\sqrt{4 \times 2} = 2\sqrt{2}\). Thus, the expression becomes \(\frac{1}{2\sqrt{2}}\).

Rationalize the Denominator

To eliminate the square root in the denominator, we multiply both the numerator and the denominator by \(\sqrt{2}\). This gives us \(\frac{1 \cdot \sqrt{2}}{2 \sqrt{2} \cdot \sqrt{2}}\).

Simplify the Fraction

Now, carry out the multiplication in the denominator: \(2 \sqrt{2} \cdot \sqrt{2} = 2 \cdot 2 = 4\). The expression now is \(\frac{\sqrt{2}}{4}\) since the numerator becomes \(\sqrt{2}\).

Key concepts

Simplifying Radicals

Simplifying radicals is a crucial step in many mathematical problems, especially when dealing with square roots. The goal here is to make the expression as simple as possible. Radicals can often be broken down into simpler components. For example, \[\sqrt{8}\] can be rewritten by identifying any perfect squares within it. Since \(8\) is equal to \(4 \times 2\) and \(4\) is a perfect square, we can take its square root: \(\sqrt{4} = 2\).
This process transforms \[\sqrt{8}\] into \[2\sqrt{2}\], making the expression simpler to work with.
Key points to remember include identifying perfect squares and separating them from other numbers under the radical. This technique makes later calculations more manageable. Additionally, simplifying radicals is an essential skill that helps in the rationalizing of denominators.

Fractions

Fractions are a way to express division between two numbers, usually represented as \( \frac{numerator}{denominator} \). They often appear in problems involving division or proportion calculations.
When dealing with fractions, always aim to simplify them as much as possible. Simplifying involves finding the greatest common factor of both the numerator and the denominator and dividing both by it.

• Another vital aspect of working with fractions is ensuring the denominator is free of radicals, which generally requires a process called 'rationalizing the denominator'.
• For example, if the denominator contains \(\sqrt{2}\), multiply both the numerator and the denominator by \(\sqrt{2}\) to eliminate the radical.

This process does not change the value of the fraction, but it simplifies it to make further calculations easier. Remember, a fraction is only fully simplified when the numerator and denominator have no common factors other than 1.

Radicals

Radicals are mathematical expressions that involve roots, such as square roots (\(\sqrt{}\)), cube roots (\(\sqrt[3]{}\)), and so on. In most cases, the simplest form of a radical expression is preferred.
Simplifying radical expressions involves finding and extracting any perfect squares present under the radical symbol. This process helps make calculations and further manipulations much more straightforward.

• To work smoothly with radicals, it's essential to understand how they interact with other numbers, especially when they appear in denominators.
• 'Rationalizing the denominator' uses techniques to eliminate radicals from the bottom of a fraction by multiplying both the numerator and denominator by a suitable radical value.

These practices provide a way to handle otherwise messy or complex expressions. Mastery of radical manipulation opens up simpler problem-solving pathways in algebra and beyond.