Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{8 b^{3}} $$

Step-by-Step Solution

Answer

The simplest radical form is \( 2b \sqrt{2b} \).

Step 1: Identify the Expression and Radicand

The expression given is \( \sqrt{8b^3} \). This is a square root, meaning we need to simplify this expression by breaking down the radicand \( 8b^3 \) into its prime factors or powers.

Step 2: Factorize the Radicand

Factorize the number 8 and the variable part separately. We have:- \( 8 = 2^3 \)- \( b^3 \) is already a power.This gives us the factors of the radicand: \( 2^3 \times b^3 \).

Step 3: Simplify Factors Inside the Radical

Apply the square root to each part under the radical separately. Remember that the square root of a power that is equal or greater than 2 means you can 'take out' a complete square:- \( \sqrt{2^3} = \sqrt{2^2 \times 2} = 2 \times \sqrt{2} \)- \( \sqrt{b^3} = \sqrt{b^2 \times b} = b \times \sqrt{b} \)

Step 4: Write the Simplified Expression

Combine these results into the simplest radical form:\( \sqrt{8b^3} = 2b \sqrt{2b} \).

Step 5: Conclusion: Verify and Box the Result

Check that all parts under the radical are in their simplest form, and the largest square factor has been taken out. Thus, the simplest radical form of the expression is confirmed to be:\( 2b \sqrt{2b} \).

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
For example, the number 8 can be broken down into its prime factors, which are 2. Specifically, 8 can be expressed as \(2 \times 2 \times 2\) or \(2^3\).

• Prime factorization helps in simplifying radicals as it allows us to express numbers in a form that can be easily broken down.
• Understanding how to break down numbers and expressions into their prime components is a foundational step in simplifying radicals.

Square Root Simplification

Simplifying square roots involves breaking down a radicand into its prime factors and then simplifying these factors.
For instance, if a radicand like \(8b^3\) is given, it is essential to express it in terms of its prime factors. We have:

• \(8 = 2^3\)
• \(b^3 = b^2 \times b\)

These can then be treated separately under the square root.
Next, apply the rule of square roots, where \(\sqrt{a^2} = a\). This allows us to separate the perfect square from those that are not:

• \(\sqrt{2^3} = \sqrt{2^2 \times 2} = 2\sqrt{2}\)
• \(\sqrt{b^3} = \sqrt{b^2 \times b} = b\sqrt{b}\)

By using these steps, any expression under the square root can be simplified efficiently.

Radicals

Radicals are mathematical expressions that represent the root of something. The most common radical is the square root, often shown by the symbol \(\sqrt{}\).
Radicals can be used for various roots, not just square roots. However, the principles of simplification remain similar across different types of roots.

• To simplify a radical, you need to identify and factor the radicand (the value under the radical sign).
• This involves using techniques like prime factorization to make simplification possible.

By expressing the radicand in its simplest form, it's easier to work with the radical. This is particularly important in algebra, where radicals are often involved in solvers and equation simplification.

Exponents in Radicals

Exponents play an essential role when working with radicals, especially in determining how to simplify them.
When you encounter a radical with an exponent, such as \(b^3\), handling the exponent correctly can lead to simplifying the radical expression.

• In radicals, variables with exponents need special attention because any exponent greater than or equal to the index of the root can be simplified.
• For instance, \(b^3\) can be split into \(b^2 \times b\), where \(b^2\) is a perfect square that can be taken out of the square root as \(b\).

Handling variables with exponents under a radical involves breaking them down in a way that matches the index, ensuring that any perfect power can be "brought out" of the radical. This technique is essential for simplifying complex expressions involving radicals with exponents.