Problem 156 - Elementary Algebra

Question

In the following exercises, simplify. $$ 7 \sqrt{d}+2 \sqrt{d} $$

Step-by-Step Solution

Answer

The simplified form is \(9 \sqrt{d}\).

Step 1: Identify Like Terms

Recognize that both terms, \(7 \, \text{and} \, 2\), contain the same radical expression \(\sqrt{d}\). Thus, they are like terms and can be combined.

Step 2: Combine Like Terms

Sum the coefficients of the like terms. Add \(7\) and \(2\) to get \(9\).

Step 3: Rewrite the Expression

Combine the result with the common radical expression to obtain \(9 \sqrt{d}\).

Key Concepts

Like Terms

In algebra, 'like terms' are terms that have identical variables raised to the same power. This similarity allows them to be combined. For example, in the expression \( 7 \sqrt{d} + 2 \sqrt{d} \), the terms \( 7 \sqrt{d} \) and \( 2 \sqrt{d} \) are like terms because they both involve the square root of \( d \). Identifying like terms is crucial for simplifying expressions. The variables and their exponents must match exactly. In this case, only the numerical coefficients (the numbers in front of the variable terms) will differ.

Combining Coefficients

Once like terms are identified, the next step is combining the coefficients. The coefficient is the numerical part of each term. For the expression \( 7 \sqrt{d} + 2 \sqrt{d} \), the coefficients are 7 and 2. To combine these, simply add them together: \( 7 + 2 = 9 \). This works because the radical part (\sqrt{d}) is the same in both terms. The result of adding the coefficients is then used to form a new term: \( 9 \sqrt{d} \). This is simpler than the original expression, but still represents the same value.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form. This involves combining like terms and performing arithmetic operations where possible. Let's look at our example \( 7 \sqrt{d} + 2 \sqrt{d} \). After identifying like terms and combining their coefficients, the simplified expression becomes \( 9 \sqrt{d} \). Simplifying expressions makes them easier to work with in further calculations. This is especially important in algebra where more complex manipulations often depend on having expressions in their simplest form.