Problem 48 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ \sqrt{2}(\sqrt{15}-\sqrt{35}) $$

Step-by-Step Solution

Answer

\(\sqrt{30} - \sqrt{70}\)

Step 1: Use the distributive property

To begin, use the distributive property to distribute \(\sqrt{2}\) across each term inside the parentheses. This means you will multiply \(\sqrt{2}\) by both \(\sqrt{15}\) and \(-\sqrt{35}\).

Step 2: Multiply square roots

When multiplying square roots, multiply the numbers inside the square roots together. This gives: \(\sqrt{2} \times \sqrt{15} = \sqrt{2 \times 15}\), and \(-\sqrt{2} \times \sqrt{35} = -\sqrt{2 \times 35}\).

Step 3: Simplify under the square root

Calculate the products under the square roots: \(2 \times 15 = 30\) and \(2 \times 35 = 70\). Thus, we have \(\sqrt{30} - \sqrt{70}\).

Step 4: Attempt to simplify further

Check if \(\sqrt{30}\) or \(\sqrt{70}\) can be simplified by looking for perfect square factors. Neither 30 nor 70 contains a perfect square other than 1, so they cannot be simplified further.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions involving multiplication across addition or subtraction. In simple terms, when you have a multiplication outside parentheses, you need to distribute or "spread" that multiplication to every term inside the parentheses.

For example, in the expression \(a(b + c)\), the distributive property tells us that it's equivalent to \(a \times b + a \times c\).

• It can be particularly handy when working with variables or more complex terms inside the parentheses.
• Always remember to keep the operation inside the parentheses intact. Like subtraction or addition.
• This property helps break down complex processes into simpler ones by handling one piece at a time.

In our exercise, we used the distributive property to multiply \(\sqrt{2}\) with each term in \(\sqrt{15}-\sqrt{35}\), resulting in \(\sqrt{2} \times \sqrt{15}\) and \(-\sqrt{2} \times \sqrt{35}\). A neat way to handle multiplication involving square roots!

Square Roots

Square roots are symbols used to find a number that, when multiplied by itself, equals the target number. The square root of a number \(x\) is written as \(\sqrt{x}\).

There are some key properties when multiplying square roots:

• The product of two square roots \(\sqrt{a} \times \sqrt{b}\) is the square root of the product of the numbers inside: \(\sqrt{a \times b}\).
• Square roots always result in positive values if we're considering real numbers.
• Root terms can often seem complex, but when split into individual parts, they follow straightforward arithmetic rules.

In our example, \(\sqrt{2} \times \sqrt{15}\) is simplified to \(\sqrt{30}\). Similarly, \(-\sqrt{2} \times \sqrt{35}\) becomes \(-\sqrt{70}\). Remember, the process involves merely re-structuring the terms under a single square root.

Simplifying Expressions

Simplifying expressions is about reducing an expression to a simpler, often more readable form, without changing its value. This can involve

• combining like terms,
• eliminating parentheses with the distributive property,
• and finding simpler representations of complex terms.

For our exercise, the expression \(\sqrt{30} - \sqrt{70}\) is already relatively simple, but simplification could involve checking for potential factors.

Once you perform the multiplication, some terms might cancel each other out or break down further. Be sure to look for opportunities to express terms in simple squares or other recognizable forms.

Perfect Square Factors

Finding perfect square factors is essential for simplifying square root expressions. A perfect square is a number like 1, 4, 9, 16, etc., where the roots are whole numbers. When simplifying a square root, aim to break it down into factors, one of which is a perfect square.

Here's how it can be helpful:

• It makes complex roots easier to calculate or approximate.
• Simplifying involves expressing the number under the square root as a product of perfect squares and other integers.

In our problem, neither 30 nor 70 has perfect square factors aside from 1.
The exercise shows that while lots of numbers can be broken down, not every result simplifies further, as some numbers remain stubbornly simple, consistent with their true values.