Problem 3 - Elementary Algebra

Question

Find each sum or difference. $$ -7 \sqrt{84 x}-12 \sqrt{189 x}+2 \sqrt{21 x} $$

Step-by-Step Solution

Answer

Answer: $-48\sqrt{21x}$

Step 1: Factor out the common factors in each term

We will start by factoring out the common factors in each term: $$ -7 \sqrt{84x} = -7 \sqrt{4*21x} = -14 \sqrt{21x} $$ $$ -12 \sqrt{189x} = -12\sqrt{9*21x} = -36\sqrt{21x} $$ $$ 2\sqrt{21x} = 2\sqrt{21x} $$ 2. Combine like terms

Step 2: Add or subtract the simplified terms

Now, we can combine the like terms to find the final answer: $$ -14\sqrt{21x} - 36\sqrt{21x} + 2\sqrt{21x} = (-14 - 36 + 2) \sqrt{21x} $$ $$ = (-48)\sqrt{21x} $$ So, the final answer is: $$ -48\sqrt{21x} $$

Key Concepts

Combining Like Terms

When working with algebraic expressions, it's common to encounter multiple terms that share the same variable parts and exponents. These are called 'like terms,' and combining them is a crucial step in simplifying algebraic expressions. To combine like terms, simply add or subtract their numerical coefficients. Imagine you have a collection of fruit, such as apples. Combining like terms is similar to grouping all the apples together to see how many you have in total.

In our exercise, the like terms are those with the same radical part, \(\sqrt{21x}\). These terms can be thought of as 'apples' that we combine, simplifying our expression to a single 'pile' of radical apples. Hence, we add or subtract their coefficients: \( -14\), \( -36\), and \( 2\), which results in \( -48\). This step simplifies the complex expression into a single radical term.

Factoring Common Factors

Factoring out common factors is a technique used to simplify expressions before performing other operations. It involves identifying and 'pulling out' the greatest common factor (GCF) from each term. Think of it as organizing a cluttered room by first removing all items that are the same and putting them together. This initial step can significantly simplify the subsequent steps of your work with the expression.

Looking at the original problem, we first recognized that each radical expression contained a factor of 21x. We then expressed 84x as \(4\times21x\) and 189x as \(9\times21x\), extracting the square roots of 4 and 9 as 2 and 3, respectively. This allowed us to reformulate the problem into a simpler version with like terms, enabling us to then combine them effectively.

Simplifying Square Roots

Simplifying square roots involves finding the prime factors of the number under the radical and 'pairing off' the primes to move them outside the radical. It's much like sorting socks; pairs come out of the drawer (the radical), while unpaired socks (prime factors) remain inside.

In our example, after factoring out common factors, we worked with \(\sqrt{21x}\), which remained unchanged as 21x cannot be further simplified by extracting square roots. In other cases, simplification would involve finding pairs of prime factors, but for 21x, since neither 21 nor x have square pairs, the expression is already in its simplest radical form.