Problem 47 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ \sqrt{7}(\sqrt{5}+\sqrt{3}) $$

Step-by-Step Solution

Answer

\( \sqrt{35} + \sqrt{21} \)

Step 1: Apply the Distributive Property

To multiply, utilize the distributive property. Distribute \( \sqrt{7} \) across the terms in the parentheses: \( \sqrt{7} \times \sqrt{5} + \sqrt{7} \times \sqrt{3} \).

Step 2: Multiply the Radicals

Multiply the radicals by recognizing that \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). So, \( \sqrt{7} \times \sqrt{5} = \sqrt{35} \) and \( \sqrt{7} \times \sqrt{3} = \sqrt{21} \).

Step 3: Combine Like Terms (if any)

Combine the results from Step 2: \( \sqrt{35} + \sqrt{21} \). Since these terms are not like terms, they cannot be combined further.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions and perform multiplications more efficiently. It states that multiplying a single term by a sum of terms can be done by distributing the multiplication to each term. In simpler terms, if you have a term outside the parentheses like in the expression \( a(b + c) \), you apply:

• Multiply \( a \) by \( b \), resulting in \( ab \).
• Multiply \( a \) by \( c \), resulting in \( ac \).

Then combine these results: \( ab + ac \). This principle helps make complex calculations more manageable and is crucial when working with expressions involving radicals, like \( \sqrt{7}(\sqrt{5}+\sqrt{3}) \). The distributive property simplifies this problem by allowing \( \sqrt{7} \) to be multiplied separately with each term inside the parentheses.

Simplifying Expressions

Simplifying expressions is the process of making them as compact and understandable as possible. When working with expressions that involve radicals, such as \( \sqrt{35} + \sqrt{21} \), simplifying involves ensuring each term is in its simplest form.
To simplify a radical expression:

• Look for perfect square factors within the radicand (the expression under the radical) that can be extracted.
• Rewrite the expression with any extracted factors outside the square root.

In our given problem, evaluate each term like \( \sqrt{35} \) and \( \sqrt{21} \) to see if they contain any perfect square factors. However, neither has perfect square factors other than 1, so they remain as \( \sqrt{35} \) and \( \sqrt{21} \) without further simplification.

Radical Multiplication

Radical multiplication involves multiplying terms that contain square roots. A useful rule to remember when multiplying radicals is: \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). This property works because a square root represents a number that, when multiplied by itself, gives the original radicand.
For example, consider the radicals \( \sqrt{7} \) and \( \sqrt{5} \). By applying this multiplication rule:

• \( \sqrt{7} \times \sqrt{5} = \sqrt{35} \).
• Similarly, \( \sqrt{7} \times \sqrt{3} = \sqrt{21} \).

Using this understanding is the key to applying radical multiplication effectively, whether you're multiplying simple or more complex radical expressions. Always remember to check if the resulting radical can be simplified further.