Problem 204 - Elementary Algebra

Question

In the following exercises, simplify. $$ 5 \sqrt{27 s^{6}}+2 \sqrt{20 s^{6}} $$

Step-by-Step Solution

Answer

15 s^3 \sqrt{3} + 4 s^3 \sqrt{5}

Step 1 - Factor Inside the Radicals

Break down the numbers inside the square roots into their prime factors. \[ 5 \sqrt{27 s^6} + 2 \sqrt{20 s^6} = 5 \sqrt{3^3 \cdot s^6} + 2 \sqrt{4 \cdot 5 s^6} \]

Step 2 - Simplify the Radicals

Express the radicals in their simplified form. Note that \sqrt{a^2 b} = a \sqrt{b}. \[ 5 \sqrt{3^3 \cdot s^6} + 2 \sqrt{4 \cdot 5 s^6} = 5 \sqrt{3 \cdot (3 \cdot s^3)^2} + 2 \sqrt{4 \cdot 5 s^6} = 5 \cdot 3 s^3 \sqrt{3} + 2 \cdot 2 s^3 \sqrt{5} \]

Step 3 - Multiply the Constants

Simplify the expressions to multiply the constants.\[ 5 \cdot 3 s^3 \sqrt{3} + 2 \cdot 2 s^3 \sqrt{5} = 15 s^3 \sqrt{3} + 4 s^3 \sqrt{5} \]

Key Concepts

Radical Expressions

A radical expression includes a root symbol (√) with a number or a variable under it. For example, \(\sqrt{27 s^{6}}\) is a radical expression. Radicals can be of different types like square roots \(\sqrt{...}\), cube roots \(\sqrt[3]{...}\), etc. The most common one is the square root.
Combining radical expressions can sometimes be necessary. For example, if we have \(5 \sqrt{27 s^{6}}+2 \sqrt{20 s^{6}}\), we need to understand how to manipulate these expressions correctly.
Understanding radicals is essential for simplifying various algebraic expressions, as they often appear in equations and functions.

Simplifying Radicals

To simplify radicals, we first need to factor the expressions inside the root. This means breaking the number or the variable under the square root into smaller parts. For instance, in the problem \( 5 \sqrt{27 s^6} + 2 \sqrt{20 s^6} \), we factored \( \sqrt{27 s^6} \) into \( \sqrt{3^3 \cdot s^6} \) and \( \sqrt{20 s^6} \) into \( \sqrt{4 \cdot 5 s^6} \).
Next, express these factors in a simplified form. Use the property \( \sqrt{a^2 b} = a \sqrt{b} \). This allows us to pull out squares from under the root. Thus, \( 5 \sqrt{3^3 \cdot s^6} = 5 \sqrt{3 \cdot (3 \cdot s^3)^2} = 5 \cdot 3 s^3 \sqrt{3} \) and \( 2 \sqrt{4 \cdot 5 s^6} = 2 \cdot 2 s^3 \sqrt{5} \).
Simplifying radicals helps to reduce the expression to its simplest form, making it easier to combine and solve.

Algebraic Multiplication

When simplifying expressions, algebraic multiplication often comes into play. This means multiplying numbers and variables by following the laws of arithmetic and algebra. For instance, after simplifying the radicals, we had \( 5 \cdot 3 s^3 \sqrt{3} + 2 \cdot 2 s^3 \sqrt{5}\). Here, we multiply the constants and combine the similar parts which gives us \( 15 s^3 \sqrt{3} + 4 s^3 \sqrt{5} \).
It's important to keep track of each multiplication step to ensure no mistakes are made. In more complicated expressions, it might help to write down each step methodically. The correct handling of algebraic multiplication ensures the expressions remain accurate and are easily understood.