Problem 1 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{12}$$

Step-by-Step Solution

Answer

\( \sqrt{12} = 2\sqrt{3} \).

Step 1: Identify Perfect Squares

The first step in simplifying a square root is to identify the perfect square factors of the number under the radical. For \( \sqrt{12} \), we can break down 12 into its prime factors: \( 12 = 2 \times 2 \times 3 = 4 \times 3 \). Notice that 4 is a perfect square (\(2^2\)).

Step 2: Separate the Radical Expression

Next, express the square root of the product in terms of the product of two square roots. For \( \sqrt{12} \), we can write it as \( \sqrt{4} \times \sqrt{3} \).

Step 3: Simplify the Perfect Square

Simplify the square root of the perfect square number. \( \sqrt{4} = 2 \), which allows us to rewrite the expression: \( \sqrt{12} = 2 \times \sqrt{3} \).

Step 4: Write the Simplified Expression

Combine the simplified part with the remaining radical. Thus, \( \sqrt{12} = 2\sqrt{3} \). This is fully simplified as there are no perfect square factors left under the radical.

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics that you will encounter often. The square root of a number is a value that, when multiplied by itself, gives the original number. It's usually represented by the radical symbol \( \sqrt{} \). For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).

When you see a problem like \( \sqrt{12} \), it's asking for a number that, multiplied by itself, equals 12. However, many numbers don't have perfect square roots as whole numbers, so we simplify them to make calculations easier. This involves looking for perfect square factors within the number.

Perfect Squares

Perfect squares are numbers that are the square of an integer. This means they result from multiplying a whole number by itself. Some common examples are 1, 4, 9, 16, 25, etc. Recognizing these is crucial when simplifying square roots.

In our exercise with \( \sqrt{12} \), we looked for the largest perfect square factor within 12. That was 4, because \( 4 = 2^2 \). Identifying perfect squares helps break down larger numbers into simpler parts. This step allows you to simplify expressions by "taking out" these perfect squares from under the square root.

Prime Factorization

Prime factorization is the method of breaking down a number into its basic building blocks- the prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself, like 2, 3, 5, 7, etc.

For \( \sqrt{12} \), we used prime factorization to express 12 as \( 2 \times 2 \times 3 \). By grouping these primes, we identified 4 as a perfect square (\( 2 \times 2 \)). Recognizing these groups helps to simplify expressions by reducing what's under the radical sign.

Simplifying Expressions

Simplifying expressions is about making them as straightforward as possible. When it comes to square roots, this means reducing the expression by extracting perfect squares from under the radical.

For \( \sqrt{12} \), we identified 4 as the perfect square and rewrote the expression as \( \sqrt{4} \times \sqrt{3} \). Then, since \( \sqrt{4} = 2 \), we simplified the expression to \( 2\sqrt{3} \). This is the most reduced form. Simplifying makes expressions easier to work with and helps in solving equations where square roots are involved.