Problem 7 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. \(3 \sqrt{20}-\sqrt{5}-2 \sqrt{45}\)

Step-by-Step Solution

Answer

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

Step 1: Simplify each square root

Begin by simplifying each square root expression. For \( \sqrt{20} \), notice that \( 20 = 4 \times 5 \), so \( \sqrt{20} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \). For \( \sqrt{45} \), since \( 45 = 9 \times 5 \), this becomes \( \sqrt{45} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \).

Step 2: Substitute simplified square roots

Substitute these simplified forms back into the expression: \( 3 \sqrt{20} - \sqrt{5} - 2 \sqrt{45} \) becomes \( 3(2\sqrt{5}) - \sqrt{5} - 2(3\sqrt{5}) \).

Step 3: Apply the distributive property

Apply the distributive property in the expression. Then we have \( 3 \times 2\sqrt{5} = 6\sqrt{5} \) and \( 2 \times 3\sqrt{5} = 6\sqrt{5} \). The expression is now \( 6\sqrt{5} - \sqrt{5} - 6\sqrt{5} \).

Step 4: Combine like terms

Combine like terms by adding and subtracting the coefficients of \( \sqrt{5} \). We perform \( 6\sqrt{5} - 6\sqrt{5} - \sqrt{5} = 0\sqrt{5} - \sqrt{5} = -\sqrt{5} \).

Step 5: Final expression

The simplified expression is \( -\sqrt{5} \). This cannot be simplified further.

Key Concepts

Simplifying Square Roots

When simplifying square roots, the goal is to break down the number inside the square root into its prime factors, or a combination of numbers where one of them is a perfect square. This process helps in rewriting the square root in a simpler, more manageable form.

Here’s how to do it:

• Identify if any factors of the number inside the square root are perfect squares. A perfect square is a number like 4, 9, 16, etc.
• For example, consider simplifying \( \sqrt{20} \):
  • Factor 20 as \( 4 \times 5 \).
  • Notice that 4 is a perfect square.
  • Therefore, \( \sqrt{20} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \).
• Similarly, with \( \sqrt{45} \):
  • Factor 45 as \( 9 \times 5 \).
  • 9 is a perfect square.
  • As a result, \( \sqrt{45} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \).

This simplification helps to further work with these expressions in algebraic operations.

Combine Like Terms

Combining like terms is a simple yet crucial step in simplifying algebraic expressions. It involves grouping terms that have the same variable or radical parts and then adding or subtracting their coefficients.

For the expression \( 6\sqrt{5} - \sqrt{5} - 6\sqrt{5} \):

• Identify "like terms" - in this case, all terms have \( \sqrt{5} \) as their radical part.
• Add and subtract the coefficients:
  • Combine \( 6\sqrt{5} \) and \(-6\sqrt{5} \) to get \( 0\sqrt{5} \), as these cancel each other out.
  • Then, calculate \( 0\sqrt{5} - \sqrt{5} \) to obtain \(-\sqrt{5} \).

Remember, combining like terms allows the simplification of expressions, making them more efficient to work with in future calculations or comparisons.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition and multiplication. They can frequently involve square roots, fractions, or other algebraic elements and can become complex.

Key elements in working with algebraic expressions include:

• Understanding different terms involved in expressions, such as constants, variables, and coefficients.
• Using operations and properties, like the distributive property: \( a(b + c) = ab + ac \), which helps in managing expressions within parentheses.
• For instance, in the problem \( 3(2\sqrt{5}) - \sqrt{5} - 2(3\sqrt{5}) \), the distributive property is applied by multiplying each term individually.

Algebraic expressions can be easily manipulated with practice, helping solve equations or represent more complex algebraic relationships. Conquering these expressions can empower further steps in learning mathematics.