Problem 20 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \sqrt{5}(\sqrt{6}-\sqrt{10}) $$

Step-by-Step Solution

Answer

Question: Simplify the expression: √5(√6 - √10). Answer: The simplified expression is √30 - √50.

Step 1: Distribute the square root of 5

Multiply the square root of 5 by each term inside the parentheses: $$ \sqrt{5}(\sqrt{6}-\sqrt{10}) = \sqrt{5} \cdot \sqrt{6} - \sqrt{5} \cdot \sqrt{10} $$

Step 2: Simplify the resulting expressions

Use the property that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\) to simplify both terms: $$ \sqrt{5} \cdot \sqrt{6} - \sqrt{5} \cdot \sqrt{10} = \sqrt{5 \cdot 6} - \sqrt{5 \cdot 10} $$

Step 3: Multiply the numbers inside the square roots

Calculate the values inside the square roots: $$ \sqrt{5 \cdot 6} - \sqrt{5 \cdot 10} = \sqrt{30} - \sqrt{50} $$

Step 4: Look for potential simplifications

Notice that the numbers inside the square roots, 30 and 50, do not have any perfect squares as factors. Therefore, we cannot simplify these square roots any further.

Step 5: Final Answer

The simplified expression is: $$ \sqrt{30} - \sqrt{50} $$

Key Concepts

Distributive Property

The distributive property can be a lifesaver when you're faced with algebraic expressions that involve both multiplication and addition or subtraction. Just like distributing brochures to houses, in math, you distribute, or multiply, a single term across terms within parentheses. While dealing with square root expressions, the distributive property allows us to multiply a square root across terms being added or subtracted inside of parentheses.

For example, when you have an expression like the one in the original exercise \( \sqrt{5}(\sqrt{6}-\sqrt{10}) \), the square root of 5 needs to be distributed, or multiplied, by each square root within the parentheses. This breaks down the problem into simpler parts, making it easier to handle complex square root expressions. Remember, always ensure that you multiply before applying any addition or subtraction.

Square Root Properties

Understanding square root properties is crucial when you want to simplify radical expressions effectively. A square root, represented as \( \sqrt{a} \), is essentially asking, 'what number multiplied by itself gives me \(a\)?' One of the most important properties for simplification is that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).

This means that if you're multiplying two square roots, you can easily combine them under a single square root sign. However, be cautious! This only works when dealing with multiplication. You cannot use this property to combine square roots if they are being added or subtracted. This property was used in Step 2 of our original exercise to create a combined square root for each product.

Multiplying Square Roots

When you multiply square roots, you're bringing them together under a single roof! The process is straightforward: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\). But, it's important to note that \(a\) and \(b\) need to be non-negative numbers, since square roots of negative numbers lead us into the realm of complex numbers.

Let's go back to our exercise for a moment. By applying this rule to Step 2 of the solution process, we combined the square roots of 5 and 6, and then 5 and 10, into single square root expressions. This is how we arrived at \(\sqrt{30}\) and \(\sqrt{50}\). Now, if these resulted in perfect squares, we could simplify them even further. But since they don't, we've gone as far as we can with multiplication.

Radical Expressions

Radical expressions are expressions that contain a root, with square roots being one of the most common types. Expressions under the square root sign are called radicands. Simplifying radical expressions usually involves eliminating any perfect square factors the radicands may have. When the radicands do not have any perfect square factors, as seen in the last step of our exercise \(\sqrt{30} - \sqrt{50}\), there's no further simplification possible.

However, if we did have perfect squares within the radicand, we could break them down. For instance, if the original exercise had contained \(\sqrt{36}\), we know that 36 is a perfect square, and thus \(\sqrt{36} = 6\). But keep an eye out; expressions like \(\sqrt{30}\) and \(\sqrt{50}\) are already in their simplest form because 30 and 50 do not have perfect square factors.