Problem 58 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ \sqrt{5}(6-\sqrt{5}) $$

Step-by-Step Solution

Answer

The simplified expression is \( 6\sqrt{5} - 5 \).

Step 1: Distribute Square Root Term

To solve the expression \( \sqrt{5}(6-\sqrt{5}) \), we start by distributing the \( \sqrt{5} \) to both terms inside the parentheses.

Step 2: Multiply Each Term

First, multiply \( \sqrt{5} \) by 6, which gives \( 6\sqrt{5} \). Next, multiply \( \sqrt{5} \) by \( -\sqrt{5} \), resulting in \( -\sqrt{5} \times \sqrt{5} = -5 \).

Step 3: Write the Expression

Combine the results from Step 2 to form the expression: \( 6\sqrt{5} - 5 \).

Step 4: Simplify

Check if the expression \( 6\sqrt{5} - 5 \) can be further simplified. Since no further simplification is possible, this is the final simplified expression.

Key Concepts

Distributive Property

The distributive property is a fundamental aspect of mathematics that allows us to simplify expressions and solve equations. In essence, when multiplying a single term by a group of terms inside parentheses, the distributive property states that you distribute, or "hand out," the single term to each term inside the parentheses. In our given problem, \( \sqrt{5}(6-\sqrt{5}) \), the \( \sqrt{5} \)must be distributed to both \( 6 \)and\( -\sqrt{5} \).

• First, multiply \( \sqrt{5} \)by \( 6 \), giving you \( 6\sqrt{5} \).
• Second, multiply \( \sqrt{5} \)by \( -\sqrt{5} \), resulting in \( -5 \).This is because\( \sqrt{5} \times \sqrt{5} = 5 \) and you apply the negative.

These two products are then combined to form the expression \( 6\sqrt{5} - 5 \).
The distributive property helps in transforming the multiplication into a simpler addition and subtraction of terms, making the expression easier to work with.

Simplifying Expressions

Simplifying expressions is about making mathematical expressions as clean and concise as possible. After distributing in our exercise, we obtain the expression \( 6\sqrt{5} - 5 \).
The task then is to tell if further simplification can be done. In this context, simplifying means looking for common factors or like terms that can be combined.

• The term\( 6\sqrt{5} \)stands alone as it is a product of a whole number and a radical.
• The \( -5 \)is a whole number, thus not combinable in this instance with the radical part.

By thoroughly inspecting each term separately, you determine that \( 6\sqrt{5} - 5 \)is the expression's simplest form since no terms can be combined or further reduced.

Mathematical Operations

Understanding basic mathematical operations is crucial for dealing with expressions involving radicals like in our exercise \( \sqrt{5}(6-\sqrt{5}) \). These operations often involve:

• **Multiplication:** As seen, multiplying radicals follows the same rules as regular numbers.\( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \) applies unless the numbers are identical, like \( \sqrt{5} \times \sqrt{5} = 5 \).
• **Addition/Subtraction:** We also have to consider these operations when combining terms. Radicals often can't be added or subtracted directly unless they're similar radicals, which\( \sqrt{5} \)terms are not with integers.

Through combining multiplication and proper handling of all terms, the final form of an expression can be found. Understanding these operations is key in breaking down and constructing expressions correctly, ensuring every term is accounted for and placed correctly in their simplest form. These basics are essential in a wide array of algebraic problems.