Problem 8 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \sqrt{a}-7 \sqrt{a}-7 $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\sqrt{a}-7\sqrt{a}-7$$. Answer: The simplified expression is $$-6\sqrt{a} - 7$$.

Step 1: Identify like terms

Since there are both constant terms and terms with square roots, we should group the terms with square roots together. The given expression can be rewritten as: $$ (\sqrt{a}-7\sqrt{a}) - 7. $$

Step 2: Combine like terms

Now, we combine the terms with square roots, by "adding" them together. Note that their coefficients follow normal rules of addition and subtraction: $$ (\sqrt{a}-7\sqrt{a}) - 7 = (1-7)\sqrt{a} - 7. $$

Step 3: Simplify the coefficients

Let's subtract the coefficients of the square root terms and simplify the expression: $$ (1-7)\sqrt{a} - 7 = -6\sqrt{a} - 7. $$ The simplified expression is: $$ -6\sqrt{a} - 7. $$

Key Concepts

Like terms in Expressions

In mathematics, "like terms" are terms that have the same variables raised to the same powers. In the given expression, both \( \sqrt{a} \) and \(-7\sqrt{a} \) are like terms because they share the same variable, \(a\), under the square root symbol. This means they can be combined.
Understanding like terms makes it easier to simplify expressions. Instead of dealing with complicated terms separately, you group them, reducing the clutter and making calculations easier to follow. Consider when you have like terms:

• The variables and their exponents must match.
• Only coefficients are different, which may be added or subtracted.

In our exercise, knowing \( \sqrt{a} \) and \(-7\sqrt{a} \) are like terms allows us to simplify by just addressing their coefficients. With practice, spotting like terms becomes quick and intuitive, aiding greatly in algebraic simplifications.

Understanding Square Roots

A square root is a number that produces a specified quantity when multiplied by itself. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). In algebra, square roots provide a way to handle non-linear terms that involve variables and exponents.
Square roots are represented using the radical symbol, \(\sqrt{ }\), and often need simplification in algebraic expressions. When simplifying, keep these points in mind:

• \(\sqrt{x^2} = x\), assuming x is positive or zero.
• You can only directly combine square roots of the same radicand.

In the exercise, \(\sqrt{a}\) and \(-7\sqrt{a}\) shared the radicand, making them combinable just like regular numbers. Handling square roots this way allows for simplification, revealing clearer pathways to solutions in algebraic contexts.

Coefficients in Algebra

Coefficients are the numerical factors that multiply the variables or expressions in algebra. When we look at the expression from the exercise, coefficients connect directly to the process of simplifying the terms:

• For \(\sqrt{a}\), the coefficient is 1 (often implied).
• For \(-7\sqrt{a}\), the coefficient is -7.

These coefficients indicate how many times the accompanying term appears. To simplify the expression, we need to manipulate these coefficients directly. In this exercise, the calculation was:\[(1 - 7) \sqrt{a} = -6 \sqrt{a}\]
Thus, coefficients play a crucial role in combining like terms, as they directly affect the arithmetic steps needed to simplify expressions. By accounting for coefficients, solving and understanding algebraic expressions becomes more straightforward and logical.