Problem 8 - Prealgebra

Question

Combine by applying the distributive property. Assume all variables represent positive numbers. $$9 \sqrt{7}+\sqrt{7}$$

Step-by-Step Solution

Answer

The expression simplifies to \(10\sqrt{7}\).

Step 1: Identify Like Terms

We start by identifying terms in the expression \(9\sqrt{7} + \sqrt{7}\). Both terms, \(9\sqrt{7}\) and \(\sqrt{7}\), have the same radical part, \(\sqrt{7}\). Since they have the same radical, we can combine them using the distributive property.

Step 2: Apply the Distributive Property

The distributive property states that \(a\cdot b + a \cdot c = a\cdot(b+c)\). In our expression, \(\sqrt{7}\) is common in both terms, so we factor it out: \(9\sqrt{7}+1\sqrt{7} = \sqrt{7}(9+1)\).

Step 3: Simplify the Expression

Now, simplify the expression inside the parentheses: \(9 + 1 = 10\). Our expression then becomes \(\sqrt{7} \cdot 10\).

Step 4: Write the Final Answer

Multiply \(\sqrt{7}\) by \(10\) to obtain the final result. The expression simplifies to \(10\sqrt{7}\).

Key Concepts

Simplifying Expressions

When we simplify expressions in algebra, our goal is to make them more compact and easier to work with. This involves breaking down expressions into simpler or more familiar forms without changing their value. In our exercise, simplification started with identifying like terms, which involves recognizing terms that can be combined together. By simplifying, we often aim to reduce clutter in the expression, so the solution process becomes clearer.

For example, consider the expression given:

• We have two terms: \(9\sqrt{7}\) and \(\sqrt{7}\).
• Both terms include the same radical component, \(\sqrt{7}\).
• By recognizing that these terms are like terms because of this common component, simplification becomes possible using the distributive property.

The accomplishment of simplifying in this case not only makes the expression easier to manage, but it also highlights the efficiency of exploiting properties such as the distributive property for mathematical simplification.

Combining Like Terms

Combining like terms is a fundamental skill in simplifying algebraic expressions. Like terms are terms that have exactly the same variable parts, each raised to the same power, thus making them directly combinable through basic arithmetic operations. In our example, the terms \(9\sqrt{7}\) and \(\sqrt{7}\) are like terms because both involve the variable component \(\sqrt{7}\).

Here's how the process works:

• Identify all terms that share the same variable parts.
• Sum or subtract the numerical coefficients of these terms while keeping the variable part unchanged.
• In the given example, combining these terms involves adding the coefficients: \(9 + 1\), resulting in a single term \(10\sqrt{7}\).

The act of combining like terms makes expressions more concise and easier to interpret, which is essential for further operations in algebra.

Radicals in Algebra

Radicals, such as square roots, frequently appear in algebraic expressions and can seem daunting at first. But understanding how to manage them is crucial in algebra. A radical expression contains a root symbol (like \(\sqrt{}\)).

In algebra, radicals can be treated much like regular variables when it comes to the processes of addition, subtraction, and even multiplication or division, provided they are of like kind:

• When simplifying expressions with radicals, explore opportunities to factor out common radical parts, just as in the problem \(9\sqrt{7} + \sqrt{7}\) where \(\sqrt{7}\) was factored out.
• Knowing that radicals adhere to properties such as the product and quotient rules helps in manipulating them like other algebraic terms.
• It's crucial to ensure terms have the same radical element to be combined.

Mastering radicals involves both combinatorial skills like combining like terms and recognizing how these terms interact with the operation rules in algebra.