Problem 21 - Prealgebra

Question

Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$4 \sqrt{20 x^{3}}+3 \sqrt{45 x^{3}}$$

Step-by-Step Solution

Answer

The expression simplifies to \( 17x\sqrt{5x} \).

Step 1: Simplifying Individual Square Roots

First, simplify each square root separately. Let's start with \( \sqrt{20x^3} \). The number 20 can be factored into prime factors: 20 = 2 * 2 * 5 = 2^2 * 5. Therefore, \( \sqrt{20x^3} = \sqrt{2^2 \cdot 5 \cdot x^3} \). Extracting the square terms, we have \( \sqrt{2^2} = 2 \) and \( \sqrt{x^3} = x^{1.5} = x\sqrt{x} \). Putting it together, \( \sqrt{20x^3} = 2x\sqrt{5x} \).

Step 2: Simplifying the Second Square Root

Next, let's simplify \( \sqrt{45x^3} \). The number 45 can be factored into prime factors: 45 = 3 * 3 * 5 = 3^2 * 5. Therefore, \( \sqrt{45x^3} = \sqrt{3^2 \cdot 5 \cdot x^3} \). Extracting the square terms, we have \( \sqrt{3^2} = 3 \) and \( \sqrt{x^3} = x^{1.5} = x \sqrt{x} \). Putting it together, \( \sqrt{45x^3} = 3x\sqrt{5x} \).

Step 3: Substituting Simplified Terms

Now, substitute the simplified square roots back into the original expression: \[ 4 \sqrt{20x^3} + 3 \sqrt{45x^3} = 4(2x\sqrt{5x}) + 3(3x\sqrt{5x}) \] which simplifies to: \[ 8x\sqrt{5x} + 9x\sqrt{5x} \].

Step 4: Combining Like Terms

Since both terms contain \( x \sqrt{5x} \) as a factor, we can combine them. This gives us: \[ 8x\sqrt{5x} + 9x\sqrt{5x} = (8x + 9x)\sqrt{5x} = 17x\sqrt{5x} \].

Key Concepts

Prime Factorization in Square Roots

Prime factorization helps to simplify square roots by breaking down numbers into their prime factors. This method is like finding the building blocks of a number, where each block is a prime number that can't be further divided. To simplify a square root: - Break down the number into its prime factors. For example, the number 20 can be broken down as \( 2^2 \times 5 \). This is the same as saying 20 is made up of two twos (\(2 \times 2\)) and one five (\(5\)).- Identify pairs of the same number, since square roots "want" pairs. Each pair can come out from under the root as a single number. - Repeat for variables, treating \( x^3 \) as \( x \times x \times x \).Using this technique, we simplify expressions under the square root to factor out and simplify overall. This makes it easier to work with the expression and eventually combine like terms.

Understanding Like Terms

Like terms are terms that have identical variable parts. In the context of this exercise, look for terms that can be combined due to shared components. - Identify terms with the same variable and root component, such as \( x \sqrt{5x} \).- Combine the coefficients (the numbers in front of these terms) just like you would with simple algebraic terms.In the exercise, both \( 8x \sqrt{5x} \) and \( 9x \sqrt{5x} \) share \( x \sqrt{5x} \), which makes them like terms. Thus, you add the coefficients: \( 8 + 9 \), keeping the shared component \( x \sqrt{5x} \) the same. This concept allows you to combine terms, leading to a simpler expression: \( 17x \sqrt{5x} \).

Square Root Properties for Simplification

Square root properties are handy when simplifying square root expressions:- **Product Property**: \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). This tells us that breaking down the root into simpler parts is possible.- **Power Property**: When dealing with powers, \( \sqrt{x^n} = x^{n/2} \), lets you simplify roots with variables. For example, \( \sqrt{x^3} = x^{1.5} = x \sqrt{x} \).Understanding these properties allows you to handle expressions under the square root confidently. Simplifying each expression separately and using square root properties wisely makes combining and further simplifying the expression much easier. By correctly applying these properties, the seemingly complex task of simplifying square roots becomes systematic and approachable.