Problem 43 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt{5}(\sqrt{5}+\sqrt{15}) $$

Step-by-Step Solution

Answer

The simplified expression is \( 5 + 5\sqrt{3} \)

Step 1: Distribute the square root

Distribute \( \sqrt{5} \) into the parenthesis to each term: \( \sqrt{5} * \sqrt{5} + \sqrt{5} * \sqrt{15} \)

Step 2: Simplify the multiplication

Use the rule \( \sqrt{a} * \sqrt{b} = \sqrt{a*b} \) to simplify each term: \( \sqrt{5 * 5} + \sqrt{5 * 15} = \sqrt{25} + \sqrt{75} \)

Step 3: Simplify the radicals

Because \( \sqrt{25} \) is a perfect square, it can be simplified to 5. For the \( \sqrt{75} \), it can be simplified by factoring 75 into 25 and 3, and then taking the square root of 25: \( 5 + \sqrt{25 * 3} = 5 + \sqrt{25} * \sqrt{3} = 5 + 5\sqrt{3} \)

Key Concepts

Simplifying Radicals

Simplifying radicals is a common task in algebra that involves making expressions containing square roots easier to work with. When you simplify a radical, you are aiming to transform it into a form where all possible factors have been broken down. This often results in fewer numbers under the radical sign, making the whole expression simpler.

• Consider the expression under the radical sign and look for perfect square factors. Perfect squares such as 4, 9, 16, and 25 simplify to 2, 3, 4, and 5 respectively, because those are their square roots.
• Break down the number under the radical into its prime factors. You can then regroup these factors into perfect squares, simplifying the radical further.
• For example, if you have the radical \( \sqrt{75} \), it's beneficial to notice that 75 can be factored into \( 25 \times 3 \), allowing you to simplify \( \sqrt{75} \) into \( 5\sqrt{3} \).

This step by step simplification makes expressions much clearer and often represents an important part of solving algebraic problems.

Distributive Property

The distributive property is a handy algebraic tool used to simplify expressions, especially those involving parentheses. This property helps in expanding expressions and solving equations quickly and systematically. The rule states that for any numbers \( a, b, \) and \( c \), the equation \( a(b + c) = ab + ac \) holds true.

• Using the distributive property means you multiply the factor outside the parentheses by each term inside the parentheses separately.
• As in the original exercise, we multiply \( \sqrt{5} \) by each term inside the parentheses, resulting in \( \sqrt{5} \times \sqrt{5} + \sqrt{5} \times \sqrt{15} \).
• This step helps in breaking down the expression into smaller parts that can be addressed individually, making the computation process more straightforward.

The distributive property allows you to manage and manipulate expressions in algebra more effectively by reorganizing terms and making simplification processes easier.

Square Roots

Square roots appear in numerous mathematical expressions, and understanding how to work with them helps in simplifying various problems, including the ones involving radicals. A square root of a number \( a \) is a value \( x \) such that \( x^2 = a \).

• Familiarize yourself with the concept of a perfect square and the idea that the square root of a perfect square is an integer. For example, the square root of 25 is 5 because 5 times 5 equals 25.
• When dealing with non-perfect squares, the square root will often be left in radical form, such as \( \sqrt{3} \), unless further simplification is possible.
• In algebra, combining or simplifying terms involving square roots often requires the operation of multiplying square roots, as seen in radicals like \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \).

Mastering the manipulation of square roots allows you to simplify complicated expressions and solve equations with confidence and ease.