Problem 25 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{12}-\sqrt{24}+\sqrt{48}+\sqrt{27} $$

Step-by-Step Solution

Answer

The simplest form is \(9\sqrt{3} - 2\sqrt{6}\).

Step 0: Identify Each Radical Expression

We will simplify each radical separately: \(\sqrt{12}\), \(\sqrt{24}\), \(\sqrt{48}\), and \(\sqrt{27}\).

Step 1: Simplify \(\sqrt{12}\)

First, write \(12\) as a product of its prime factors: \(12 = 2^2 \times 3\). Rewrite \(\sqrt{12}\) as \(\sqrt{2^2 \times 3} = 2\sqrt{3}\).

Step 2: Simplify \(\sqrt{24}\)

Factor \(24\) into primes: \(24 = 2^3 \times 3\). Rewrite \(\sqrt{24}\) as \(\sqrt{2^3 \times 3} = \sqrt{4 \times 6} = 2\sqrt{6}\).

Step 3: Simplify \(\sqrt{48}\)

Factor \(48\): \(48 = 2^4 \times 3\). So, \(\sqrt{48} = \sqrt{2^4 \times 3} = \sqrt{16 \times 3} = 4\sqrt{3}\).

Step 4: Simplify \(\sqrt{27}\)

Factor \(27\): \(27 = 3^3\). Therefore, \(\sqrt{27} = \sqrt{3^3} = \sqrt{9 \times 3} = 3\sqrt{3}\).

Step 5: Combine Simplified Expressions

Substitute the simplified radicals into the expression: \(2\sqrt{3} - 2\sqrt{6} + 4\sqrt{3} + 3\sqrt{3}\).

Step 6: Simplify the Expression Further

Combine like terms: \((2\sqrt{3} + 4\sqrt{3} + 3\sqrt{3}) = 9\sqrt{3}\). Therefore, \(9\sqrt{3} - 2\sqrt{6}\).

Key Concepts

Simplifying Radicals

When we talk about simplifying radicals, we're referring to the process of rewriting a square root in a simpler form. It involves reducing the expression under the square root sign, also known as the radicand, to its most basic components. This often means factoring the radicand to find perfect squares that can be "pulled out" from under the square root.
For example, simplifying \(\sqrt{12}\) involves knowing that 12 can be broken down as \(2^2 \times 3\) so we can express it as \(\sqrt{2^2 \times 3} = 2\sqrt{3}\). The factor \(2^2\) (which is \(4\)) is a perfect square, allowing us to take 2 out from the radical sign, simplifying the expression significantly.
This step-by-step approach ensures we simplify radicals down to their most reduced form, which is crucial in solving or simplifying larger algebraic expressions involving multiple radical terms.

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime numbers. This technique is key when simplifying radicals, as it allows us to see which factors hold perfect squares.
Taking a number like 24, we factor it into prime numbers: \(24 = 2^3 \times 3\). This tells us that we have three 2's and one 3 multiplying each other to make 24. With prime factorization, it becomes clear that \(4\) (or \(2^2\)) is a perfect square. It can be helpful even in more complex expressions because once we identify these perfect squares, we can "pull them out" of the square root, simplifying the expression.

• Factor each number into its prime components.
• Look for perfect squares among the factors.
• Extract the perfect squares from the square root.

Prime factorization not only helps in simplifying radicals but also ensures accuracy when housing different expressions under the same operation.

Like Terms

Like terms are terms in an expression that have the exact same variable parts raised to the same power. When simplifying radicals, identifying like terms becomes essential when combining terms.
For instance, if you have \(2\sqrt{3}\), \(4\sqrt{3}\), and \(3\sqrt{3}\), they are like terms because they all have the same radical part, \(\sqrt{3}\). This means when you combine these terms, you simply deal with the coefficients: \(2 + 4 + 3 = 9\). Therefore, they simplify to \(9\sqrt{3}\).
Recognizing like terms allows you to perform operations like addition and subtraction on them effectively, streamlining the expression into its simplest form.

Square Roots

Square roots, denoted as \(\sqrt{}\), represent a number that, when multiplied by itself, gives that number under the root. Understanding this concept lies at the heart of simplifying radical expressions.

• For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).
• Similarly, \(\sqrt{49} = 7\) as \(7 \times 7 = 49\).

In radical expressions, it's about identifying parts of the radicand that can be perfectly squared and "pulled out" from the under the root.
Sometimes the radicand isn’t a perfect square, which is where simplification and understanding perfect squares through factorization become important. Bringing this understanding into problem-solving enables students to manage and manipulate expressions efficiently, laying a foundation for more advanced algebraic concepts.