Problem 4 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ 3 \sqrt{45 x^{3}}+x \sqrt{5 x} $$

Step-by-Step Solution

Answer

The expression simplifies to \(10x \sqrt{5x}\).

Step 1: Simplify the First Term

We begin by simplifying the term \(3 \sqrt{45 x^3}\). Start with the prime factorization of 45, which is \(3^2 \times 5\). So, we have: \[3 \sqrt{45 x^3} = 3 \sqrt{3^2 \times 5 \times x^3}\]Next, we factor out perfect squares from the square root. Since \(3^2\) is a perfect square and \(x^3\) can be split into \(x^2 \times x\), we have:\[3 \times 3 \times x \times \sqrt{5x} = 9x \sqrt{5x}\]

Step 2: Simplify the Second Term

The second term is \(x \sqrt{5x}\). Since \(x\) is already outside the square root, this term is simplified as is. It remains: \[x \sqrt{5x}\]

Step 3: Combine Like Terms

Now, notice that both terms have a common factor of \(\sqrt{5x}\). This allows us to combine them: \[9x \sqrt{5x} + x \sqrt{5x} = (9x + x) \sqrt{5x} = 10x \sqrt{5x}\]

Step 4: Final Expression

Having combined the like terms, the simplified expression is: \[10x \sqrt{5x}\]

Key Concepts

Simplifying Radicals

When you come across a radical expression, the first step to simplifying it is often breaking it down into more manageable parts. This means finding any perfect squares (or other powers) within the radical so they can be taken out. Here’s how it works:

• Look for perfect squares: For example, in the expression \( \sqrt{45 x^3} \), you want to find something like \( 3^2 \) because it is a perfect square.
• Factor everything: We factor \( 45 \) into its prime components, which gives \( 3^2 \times 5 \).
• Rewrite the expression: By recognizing \( x^3 \) can be rewritten as \( x^2 \times x \), we can further simplify this term.
• Extract what's possible: Since \( 3^2 \) and \( x^2 \) are perfect squares, they can be removed from under the square root, leaving us with: \( 3 \times 3 \times x \times \sqrt{5x} \).

Simplifying radicals allows you to express the terms in their simplest forms, making it easier to spot common factors later.

Prime Factorization

Prime factorization is a process used to express a number as a product of its prime factors, which are numbers greater than 1 that have no divisors other than 1 and themselves. Understanding this is essential for simplifying radicals:

• Break it down: Start with any number you want to factor, like the 45 in \( 45x^3 \), and divide by the smallest prime number (in this case, 3) until you can’t anymore.
• Repeat with next smallest prime: After dividing by 3 a couple of times, since \( 45 = 3^2 \times 5 \), you are left with 5, which is also a prime number.

By performing prime factorization, you identify perfect squares effortlessly, making it straightforward to extract and simplify expressions like \( \sqrt{45x^3} \) into \( 9x \sqrt{5x} \). This step is crucial as reducing expressions to their minimal terms not only aids in simplification but also makes subsequent calculations much easier.

Combining Like Terms

After simplifying the expressions within a problem, the next step is often to combine like terms. This means looking for terms that share the same variable parts:

• Identify similar terms: In our original problem, once the radicals have been simplified, we see both terms present as something times \( \sqrt{5x} \).
• Factor the expression: Treat \( \sqrt{5x} \) as a common factor as you would any other variable, which allows you to add (or subtract) the coefficients, much like simple algebra.

For example, after simplifying to \( 9x \sqrt{5x} + x \sqrt{5x} \), these are both terms with \( \sqrt{5x} \) as a factor. Combine them like this:- Add the coefficients (9x and x) to get 10x.- Keep the common radical part, \( \sqrt{5x} \).The final combined expression is \( 10x \sqrt{5x} \). Combining like terms refines your answer and provides a neat, final result that is much easier to interpret.