Problem 61 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ -2 \sqrt{27} $$

Step-by-Step Solution

Answer

-6\(\sqrt{3}\).

Step 1: Prime Factorize the Radicand

The first step is to factorize our radicand, in this case, 27. 27 can be expressed as \(3^3\). Therefore our expression now looks as follows: -2\(\sqrt{3^3}\).

Step 2: Apply Radical Multiplication Rule

Let's apply the rule \(\sqrt{ab} = \sqrt{a}\sqrt{b}\) to split the expression into simpler roots: -2\(\sqrt{3^3}\) = -2*\(\sqrt{3^2} *\sqrt{3}\).

Step 3: Simplify Radicals

Reduce the indicated roots. \(\sqrt{3^2}\) becomes 3, while \(\sqrt{3}\) remains as is because 3 is a prime number. Thus, our final expression is: -2*3\(\sqrt{3}\).

Step 4: Multiply Constants

Let's now multiply the constants outside the square root, obtaining: -6\(\sqrt{3}\). This is our simplest form expression.

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that involve roots, like square roots, cube roots, or higher roots. These expressions often appear in algebra problems and can sometimes look complex at first glance. A key part of solving problems involving radical expressions is simplifying them, which involves rewriting the expression in its simplest form.

For example, take the expression \(-2 \sqrt{27}\). It's not immediately clear what the simplest form is, but by breaking it down, we can simplify it. Simplifying involves expressing the radicand, the number under the radical sign, in such a way that it makes calculations easier.

To start with simplifying a radical expression, it's crucial to know about prime factorization and radical rules, which we'll discuss in the following sections. These techniques help transform a radical into a more approachable form, where it becomes easier to handle and, consequently, solve.

• Radicals are indicated by the square root symbol (√).
• The number inside the radical is called the radicand.
• Simplification involves reducing the radicand to its simplest form.

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime numbers. Primes are numbers greater than 1 that have no divisors other than 1 and themselves. In the context of simplifying radicals, prime factorization allows us to break down the radicand into parts that make simplifying possible.

Consider the radicand in our example \(27\). We can express 27 as \(3^3\) because multiplying three 3's gives us 27. By breaking down the number to its prime factors, we pave the way to simplify the radical expression.

This step is crucial because it enables us to apply radical rules effectively, making calculations straightforward and less cumbersome. Without prime factorization, simplifying complex expressions could become a mental juggling act with too many variables to keep track of.

• Breaking down numbers into primes helps in managing them within radicals.
• Prime factorization transforms complex numbers into single-digit multipliers where possible.
• This step is the foundation for applying radical multiplication rules.

Radical Multiplication Rule

The radical multiplication rule states that the square root of a product can be expressed as the product of square roots. In symbols, this rule can be represented as \(\sqrt{ab} = \sqrt{a}\sqrt{b}\). Using this rule helps us simplify radical expressions by splitting a complicated radical into simpler parts.

For instance, after factorizing the radicand 27 into \(3^3\), the expression \(-2 \sqrt{3^3}\) can be rewritten using the rule into \(-2 \sqrt{3^2} \sqrt{3}\). Here, we separate the expression into two parts: one that we can easily simplify (\