Problem 41 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt{2}(\sqrt{50}+7) $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( 10+7 \sqrt{2} \).

Step 1: Simplify the Square Root Expressions

Begin by simplifying the square root expressions. For \( \sqrt{50} \), rewrite the number 50 as \( 25*2 \), and then simplify it to \( \sqrt{25}*\sqrt{2} = 5 \sqrt{2} \). So, the original expression now becomes \( \sqrt{2}(5\sqrt{2}+7) \).

Step 2: Distribute \( \sqrt{2} \)

Distribute \( \sqrt{2} \) to both terms within the brackets. So, \( \sqrt{2} * 5\sqrt{2} + \sqrt{2} * 7 \). Multiplying \( \sqrt{2} \) by \( 5\sqrt{2} \) equals \( 5*2 \) or 10. And \( \sqrt{2} * 7 \) becomes \( 7 \sqrt{2} \), so the expression is now \( 10+7 \sqrt{2} \).

Step 3: Write the Final Simplified Form

Upon simplifying the expression totally, the final simplified form is \( 10+\sqrt{2}*7 \), or \( 10+7 \sqrt{2} \).

Key Concepts

Square Roots

Square roots are one of the foundational concepts in mathematics. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 25 is 5 because 5 multiplied by itself equals 25. Square roots are denoted using the radical symbol \(\sqrt{}\). Understanding how to simplify square roots is crucial for various mathematical operations.

• When simplifying square roots, look for factors of the number under the square root that are perfect squares.
• For example, to simplify \(\sqrt{50}\), we find that 50 can be expressed as 25 \( * \) 2.
• Since 25 is a perfect square, \(\sqrt{25}\) becomes 5, so \(\sqrt{50}\) simplifies to 5\(\sqrt{2}\).

Simplifying square roots can help simplify expressions and solve equations more efficiently.

Rationalizing Denominators

Rationalizing the denominator is the process of eliminating any square roots or other irrational numbers from the denominator of a fraction. This is done because expressions are generally considered to be in their simplest form when the denominator is a rational number.

• To rationalize a denominator, multiply both the numerator and denominator by a suitable radical that makes the denominator a perfect square.
• For example, to rationalize the denominator of \(\frac{1}{\sqrt{2}}\), multiply both the numerator and denominator by \(\sqrt{2}\).
• This gives you \(\frac{\sqrt{2}}{2}\), which is a fraction with a rational denominator.

Rationalizing denominators not only simplifies the expression, but it also makes mathematical operations, involving the expression, clearer.

Distributive Property

The distributive property is a fundamental algebraic property that is used to simplify expressions and solve equations. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. Mathematically, this property is written as: \(a(b + c) = ab + ac\).

• The distributive property allows you to simplify expressions by distributing multipliers over grouped terms.
• For the expression \(\sqrt{2}(5\sqrt{2} + 7)\), apply the distributive property by multiplying \(\sqrt{2}\) by both 5\(\sqrt{2}\) and 7.
• This results in \(10 + 7\sqrt{2}\), showing how each term is separately multiplied by \(\sqrt{2}\).

This property is useful for breaking down complex expressions and is often used alongside other properties to simplify and solve algebraic expressions more effectively.