Problem 71 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{44}-4 \sqrt{11} $$

Step-by-Step Solution

Answer

The simplified expression is \(-2 \sqrt{11}\).

Step 1: Identify the Expression

The expression given is \( \sqrt{44} - 4 \sqrt{11} \). The goal is to simplify this expression.

Step 2: Simplify the Square Root

To simplify \( \sqrt{44} \), we need to find the factors of 44 that are perfect squares. Notice that \( 44 = 4 \times 11 \). So, \( \sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} = 2 \sqrt{11} \).

Step 3: Substitute and Combine Like Terms

We substitute back into the original expression: \( 2 \sqrt{11} - 4 \sqrt{11} \). Since these are like terms, we can subtract them: \( (2 - 4) \sqrt{11} = -2 \sqrt{11} \).

Key Concepts

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4, because 4 times 4 equals 16. Simplifying square roots involves finding a factorization of the number under the root, where at least one factor is a perfect square.

To simplify \( \sqrt{44} \), we can look for perfect square factors of 44. A perfect square is a number like 4, 9, 16—numbers which are squares of whole numbers.

- Start by breaking down 44 into its factors: \( 44 = 4 \times 11 \).
- Recognize that 4 is a perfect square, so we can use its square root.

Simplifying further, \( \sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} = 2\sqrt{11} \). This shows how roots can be simplified by separating the square root of a perfect square.

Like Terms

The concept of like terms refers to terms that have the exact same variable components raised to the same powers. In algebra, simplifying expressions often involves combining these like terms. This is because they share common elements and can thus be added or subtracted easily.

Consider the expression \( 2 \sqrt{11} - 4 \sqrt{11} \). Both terms involve \( \sqrt{11} \) as a factor, making them like terms.

- The coefficients (2 and -4) determine how many times to count \( \sqrt{11} \).
- Subtracting these coefficients: \( 2 - 4 = -2 \).

The expression simplifies to \( -2 \sqrt{11} \), because each term contains the same square root. By understanding like terms, simplifying becomes straightforward when shared factors are evident.

Factoring Perfect Squares

Factoring involves breaking down a number or an algebraic expression into its constituent elements. For perfect squares, the factors are numbers that are squares of integers.

In the context of simplifying square roots, identifying perfect squares makes the process efficient. For example, a number like 44 can be factored into 4 and 11. Here, 4 is a perfect square (as the square of 2), which simplifies the expression \( \sqrt{44} \) into \( 2\sqrt{11} \).

- Recognizing perfect squares within the factors allows us to pull them out of a square root, simplifying the expression greatly.
- This technique not only helps with numbers but also algebraic expressions where terms involve squares.

Factoring perfect squares is a crucial part of algebra, laying the groundwork for simplifying complex expressions efficiently by focusing on achievable simplifications through factorization.