Problem 14 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ -\sqrt{10}-2 \sqrt{10} $$

Step-by-Step Solution

Answer

Question: Simplify the expression: $-\sqrt{10}-2\sqrt{10}$. Answer: The simplified expression is $-3\sqrt{10}$.

Step 1: Identify like terms

In the given expression, both terms have square root of 10 as the common part. So, they can be combined together. $$ -\sqrt{10}-2\sqrt{10} $$

Step 2: Combine like terms

Now, we combine the like terms by considering the square root of 10 as a common factor and adding the coefficients. $$ (-1-2)\sqrt{10} $$

Step 3: Simplify the expression

Performing the arithmetic operation on the coefficients. $$ (-3)\sqrt{10} $$ The simplified expression is: $$ -3\sqrt{10} $$

Key Concepts

Understanding Like Terms

In algebra, like terms are terms that have the same variable raised to the same power. When simplifying expressions, identifying like terms allows you to combine them, which simplifies the expression itself. In the given exercise, both terms, \(-\sqrt{10}\) and \(-2\sqrt{10}\), involve \(\sqrt{10}\). This makes them like terms, because they both have the same square root part.When combining like terms:

• Ensure the variable and any associated roots or exponents are identical.
• Add or subtract the coefficients in front of each like term.

In our example, \(+(-1)\) and \(-2\) are the coefficients, and adding these coefficients gives us \(-3\). Recognizing and combining like terms is an essential skill for simplifying expressions smoothly.

Demystifying 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 9 is 3 because \(3 \times 3 = 9\). Square roots can also be expressed as \(x^{1/2}\) in algebra.In the expression \(-\sqrt{10}-2\sqrt{10}\), the term \(\sqrt{10}\) represents the square root of 10. When you encounter square roots in expressions:

• Note that similar square roots across different terms make those terms like terms (if their coefficients differ).
• Cube or multiply them according to the problem's requirements, particularly when keeping track of coefficients.

Understanding square roots is vital for correctly simplifying expressions and solving diverse math problems.

Importance of Coefficients

In algebraic expressions, a coefficient is a number that multiplies a variable or a root. For example, in \(-3\sqrt{10}\), \(-3\) is the coefficient of \(\sqrt{10}\).When simplifying expressions:

• The coefficients are the numbers in front of a variable or root, representing the terms' multiplicative parts.
• Arithmetic operations on coefficients are straightforward but crucial for correct simplification.

In our example, combining the coefficients \(-1\) and \(-2\) resulted in \(-3\), which simplified the expression to \(-3\sqrt{10}\). By understanding coefficients, you are able to manipulate, add, or subtract algebraic terms efficiently, leading to clear and concise solutions.