Problem 58 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ \sqrt{5}(6-\sqrt{5}) $$

Step-by-Step Solution

Answer

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

Step 1: Distribute the Square Root

We start by applying the distributive property to multiply each term in the parenthesis by the expression outside. Multiply \( \sqrt{5} \) by \( 6 \) and then \( \sqrt{5} \) by \( -\sqrt{5} \).

Step 2: Multiply First Term

Multiply \( \sqrt{5} \) by \( 6 \) to get \( 6\sqrt{5} \).

Step 3: Multiply Second Term

Multiply \( \sqrt{5} \) by \( -\sqrt{5} \). This becomes \( -\sqrt{5} \times \sqrt{5} = -\sqrt{25} = -5 \).

Step 4: Combine the Terms

Combine the results from Step 2 and Step 3. The expression becomes \( 6\sqrt{5} - 5 \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while ensuring that the value remains unchanged. It's a crucial technique in algebra that helps make expressions easier to work with. In our exercise, we use the distributive property to simplify the expression, \( \sqrt{5}(6-\sqrt{5}) \).
This property allows us to multiply each term inside the parenthesis by the term outside, in this case, \( \sqrt{5} \). It breaks down the expression step by step, ensuring that we handle each multiplication correctly.

• First, \( \sqrt{5} \) is multiplied by \( 6 \), resulting in \( 6\sqrt{5} \).
• Then, \( \sqrt{5} \) is multiplied by \( -\sqrt{5} \), giving a result of \( -\sqrt{25} \), or \( -5 \).

After calculating these products, the terms are combined to produce \( 6\sqrt{5} - 5 \), the simplified expression. By fully understanding each step of simplification, you'll have a much easier time with algebraic expressions of all kinds.

Square Roots

Square roots are essentially values that, when multiplied by themselves, yield the original number. For example, the square root of 25 is 5 because 5 multiplied by 5 equals 25. In our exercise, square roots appear both in the original expression and during the simplification process.
Understanding how to work with square roots is fundamental in many areas of math. When simplifying an expression with square roots, it is important to recognize familiar values like \( \sqrt{25} = 5 \). This makes multiplying and simplifying radicals more intuitive.

• In the expression \( -\sqrt{5} \times \sqrt{5} \), we simplify this to \( -\sqrt{25} \), which equals \(-5\).

Recognizing these values and being able to simplify them helps you manage expressions more efficiently. It not only makes the process easier but also prevents errors that can result from miscalculations.

Multiplication of Radicals

Multiplying radicals, such as square roots, follows specific rules similar to regular multiplication. When multiplying radicals, you can multiply the numbers inside the radicals together, as long as both radicals are over the same root.
In the example \( \sqrt{5} \times \sqrt{5} \), you multiply the numbers inside the radical which results in \( \sqrt{25} \), simplifying to 5.

• First, ensure that both radicals are of the same type (like square roots).
• Multiply the terms inside the radicals.
• Simplify the resulting radical if possible.

In some cases, multiplying radicals can remove the radical if the numbers under the radicals form a perfect square, as shown in the exercise. Understanding these steps will boost your confidence in handling tasks involving radicals in algebra.