Problem 21 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \sqrt{a}(\sqrt{a}-\sqrt{b c}) $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\sqrt{a}(\sqrt{a}-\sqrt{bc})\). Answer: \(a - \sqrt{abc}\)

Step 1: Apply the distributive property

We will start by applying the distributive property of multiplication to distribute the square root of a term. $$ \sqrt{a}(\sqrt{a}-\sqrt{bc}) = \sqrt{a}\cdot\sqrt{a} - \sqrt{a}\cdot\sqrt{bc} $$

Step 2: Simplify the terms

Now we will simplify the terms. For the first term, we have \(\sqrt{a}\cdot\sqrt{a}\), which is actually \(a\). For the second term, we have \(\sqrt{a}\cdot\sqrt{bc}\), which is \(\sqrt{abc}\). So, we have: $$ a - \sqrt{abc} $$ The simplified expression is: $$ a - \sqrt{abc} $$

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra and arithmetic. It allows you to multiply a single term across a sum or difference inside a set of parentheses. In simple terms, it means you distribute one term to each term inside the parentheses and perform the multiplication.

Here's a simple example: Suppose you have an expression like this: \( x(y + z) \). Applying the distributive property, you'll multiply \(x\) by both \(y\) and \(z\), so it becomes \(xy + xz\).

• An essential feature of the distributive property is that it can simplify complex expressions, making them easier to manage.
• It is especially useful for dealing with polynomial expressions and expressions involving variables.
• This property holds for real numbers, which includes integers, fractions, and irrational numbers like square roots.

When simplifying expressions using this property, do not rush. Distribute carefully and ensure each term is multiplied correctly. Practice helps in mastering this foundational concept, and once you have a good grip on it, algebraic simplifications become much easier.

Square Roots

Square roots are another essential mathematical concept often used in algebra. The square root of a number \(x\), represented by \(\sqrt{x}\), is a value that, when multiplied by itself, gives the original number \(x\).

Some important points about square roots include:

• The square root of a perfect square (like 1, 4, 9) is an integer. For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).
• For non-perfect squares, the square root will be an irrational number. For instance, \(\sqrt{2}\) is approximately 1.414, a non-repeating, non-terminating decimal.
• Square roots follow the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\), which is beneficial when simplifying expressions, as seen in the problem.

Understanding how square roots work is vital, especially when solving equations that involve quadratic terms, as it frequently appears in both algebraic expressions and real-world scenarios.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division. They form the core of algebra and are used to model real-world situations.

In algebraic expressions:

• Variables like \(x, y,\) or in our case \(a, b, c\), represent unknown values and are often solved for.
• Constants are fixed numbers that do not change.
• Operators (+, −, ×, ÷) connect variables and constants to form expressions, which can further be simplified using algebraic properties like distributive property or combining like terms.

An example would be simplifying an expression such as \(3x + 2x - 5\). Here, 3x and 2x are like terms and can be combined to \(5x - 5\).

Mastering algebraic expressions involves recognizing patterns, understanding variable manipulation, and applying algebraic rules effectively. This skill builds the foundation for solving equations and inequalities, vital steps in advancing through mathematics.