Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ (\sqrt{12})^{2} $$

Step-by-Step Solution

Answer

The simplest form is 12.

Step 1: Understand the Problem

We need to simplify the expression \((\sqrt{12})^2\). This involves finding the simplest form of a square root expression raised to the power.

Step 2: Apply the Power Rule to the Square Root

Use the property of exponents that relates powers and roots: \((\sqrt{a})^2 = a \). Therefore, \((\sqrt{12})^2\) simplifies to 12.

Key Concepts

Simplifying Expressions

Simplifying expressions means rewriting them in a way that's easier to understand or work with. It involves removing any complexities or unnecessary parts of a given mathematical expression.
In the context of radicals, this often involves reducing expressions under square roots or combining like terms effectively. To simplify, look for opportunities to break down complex terms into more manageable ones and combine them.

• Identify operations that can be simplified, such as multiplying any terms that are exponents or using distributive properties to expand expressions.
• Ensure that every step maintains the equality of the expression.
• When dealing with square roots, determine if the number can be factored into prime numbers and simplified further.

Simplifying results in cleaner, more concise expressions that are generally easier to interpret and use in subsequent calculations.

Square Root

A square root of a number is a value that, when multiplied by itself, gives the original number. It is often represented by the radical symbol \(\sqrt{}\). In mathematical equations, we frequently encounter square roots in terms of operations or simplifications.
When given an expression under a square root, like \(\sqrt{12}\), the goal is typically to simplify it. Here's how:

• Break the number into prime factors to see if any perfect squares exist.
• In the case of \(12\), it can be broken into \(4 \times 3\), where \(4\) is a perfect square.
• Thus, \(\sqrt{12}\) can be simplified to \(\sqrt{4} \times \sqrt{3} = 2\sqrt{3}\).

Always check if the expression can be further simplified. Simplifying square roots makes dealing with square roots in equations easier and helps to ensure that our results remain tidy and accessible.

Exponents

Exponents are a way of representing repeated multiplication of a number by itself. In the expression \((\text{{number}})^n\), \(n\) is the exponent. Its role is crucial when dealing with powers and roots in mathematical expressions.
A fundamental aspect of exponents is understanding the Power Rule for Radicals: if you have a square root of a number raised to the power of 2, you multiply the inner number directly. This is because \( (\sqrt{a})^2 = a \).

• This principle helps when simplifying expressions involving roots and powers.
• Exponents manage the expression's magnitude, allowing complex numbers to be reduced succinctly.

When applying exponents, always remember to respect the order of operations, and use properties of exponents to simplify whenever possible. This includes combining like bases, adjusting exponents rather than expanding unnecessarily, and using equivalent expressions to keep calculations simple and accurate.