Problem 37 - 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. $$ \sqrt{50 x^{3}}+\sqrt{200 x^{3}} $$

Step-by-Step Solution

Answer

The simplest form is \( 15x\sqrt{2x} \).

Step 1: Simplify Each Square Root

To simplify, start with each individual square root. Break down the numbers and variables into their prime factors.For \( \sqrt{50x^3} \):\[ \sqrt{50x^3} = \sqrt{25 \times 2 \times x^2 \times x} = \sqrt{25} \times \sqrt{2} \times \sqrt{x^2} \times \sqrt{x} = 5x\sqrt{2x} \]For \( \sqrt{200x^3} \):\[ \sqrt{200x^3} = \sqrt{100 \times 2 \times x^2 \times x} = \sqrt{100} \times \sqrt{2} \times \sqrt{x^2} \times \sqrt{x} = 10x\sqrt{2x} \]

Step 2: Combine Like Terms

After simplifying the square roots, you have:\( 5x\sqrt{2x} + 10x\sqrt{2x} \).Since both terms have \( x\sqrt{2x} \) as a common factor, we can combine them:\[ (5x + 10x)\sqrt{2x} = 15x\sqrt{2x} \]

Key Concepts

Square Roots

Square roots help us find a number that, when multiplied by itself, gives the original number. When faced with an expression like \( \sqrt{50x^3} + \sqrt{200x^3} \), breaking down each part into simpler components is key. For each square root:

• Identify the numbers or variables inside the root.
• Split the numbers into more manageable factors.
• Find perfect squares among them because these will easily come out of the square root sign.

For example, in \( \sqrt{50x^3} \), \( 50 \) can be split into \( 25 \times 2 \), where \( 25 \) is a perfect square. This helps in further simplifying the square root. Remember, for variables like \( x^3 \), take the even powers out first so you can easily work with them.

Like Terms

Like terms make combining parts of an expression easier. When you simplify expressions, look for terms that share the same variables raised to the same powers. For instance, in the step-by-step solution:

• Both \( 5x\sqrt{2x} \) and \( 10x\sqrt{2x} \) have the factor \( x\sqrt{2x} \).
• Since the \( \sqrt{2x} \) parts match, you treat them as similar or 'like' terms.
• Add their coefficients, which are the numbers in front (here, \( 5x \) and \( 10x \)).

This results in combining them efficiently into a single term \( 15x\sqrt{2x} \). Doing this reduces the complexity of the expression and makes further operations simpler.

Prime Factorization

Prime factorization involves expressing a number as a product of prime numbers. It's a fundamental concept to simplify square roots. To factor a number:

• Break the number down into the smallest prime numbers that multiply together to give the original number.
• Use these factors to easily identify perfect squares.
• Rewrite the original numbers in terms of these factors.

For example, 50 can be decomposed into \( 5 \times 5 \times 2 \) and 200 into \( 2 \times 2 \times 5 \times 5 \times 2 \). This representation reveals perfect squares, which can be simplified further. Prime factorization is the backbone of simplifying radicals effectively.

Simplifying Expressions

Simplifying expressions is all about reducing them to their most condensed form without changing their value. This often involves:

• Breaking down large numbers and variables into their basic components.
• Removing redundant parts of the expression.
• Making the expression as easy to interpret as possible.

Start with simplifying each part separately, using techniques like prime factorization. Then combine like terms, as seen with \( 5x\sqrt{2x} + 10x\sqrt{2x} \) transforming to \( 15x\sqrt{2x} \). Simplification makes mathematical expressions more manageable and is crucial for understanding deeper relationships between different terms in algebra.