Problem 7 - Prealgebra

Question

Combine by applying the distributive property. Assume all variables represent positive numbers. $$8 \sqrt{7}+\sqrt{7}$$

Step-by-Step Solution

Answer

The combined expression is \(9\sqrt{7}\).

Step 1: Identify Common Term

First, observe that both terms in the expression \(8\sqrt{7} + \sqrt{7}\) include the common term \(\sqrt{7}\). This term is being added to itself, making it a candidate for application of the distributive property.

Step 2: Apply the Distributive Property

By the distributive property, if we have terms like \(ab + ac\), it can be rewritten as \(a(b+c)\). Here, \(a\) is \(\sqrt{7}\), \(b\) is 8, and \(c\) is implicitly 1 since \(\sqrt{7}\) appears alone. So, apply this property to get: \(\sqrt{7}(8+1)\).

Step 3: Simplify the Expression

Calculate the sum inside the parentheses: \(8 + 1 = 9\). Thus, the expression becomes \(\sqrt{7} \times 9\), or more neatly written as \(9\sqrt{7}\).

Key Concepts

Understanding Radicals

Radicals are expressions that involve roots, typically square roots. A square root of a number is a value that, when multiplied by itself, gives the original number. For example,

• The square root of 9 is 3 because \(3 \times 3 = 9\).
• The radical symbol, \(\sqrt{}\), is used to denote square roots and other roots like cube roots, fourth roots, etc.

In our exercise, we have \(\sqrt{7}\), which represents the square root of 7. Keep in mind that a radical expression like this keeps its simplified form unless you are directed to approximate or further simplify it in a specific context.
Radicals can be thought of as a way to "undo" raising something to a power. They are a fundamental part of algebra, often appearing in various calculations.
Understanding how to manipulate radicals is essential for simplifying expressions and solving equations that involve these terms.

Simplifying Expressions

Simplifying expressions means transforming a complex mathematical expression into a simpler one without changing its value. This often makes expressions easier to work with or understand. In many cases, it involves using operations and properties like the distributive property to combine like terms or to factor expressions.
In the given problem, we are using the distributive property to simplify the expression \(8\sqrt{7} + \sqrt{7}\). The distributive property allows you to "distribute" a common factor over a sum or difference, making it possible to group and combine terms effectively. The process begins with identifying any common factors in the terms you are working with. Here, the common factor is \(\sqrt{7}\).
So, applying the distributive property:

• Identify the common term, which is \(\sqrt{7}\).
• Rewrite the expression as \(\sqrt{7}(8+1)\).
• Simplify the terms inside the parentheses, which results in \(9\), giving us the final simplified expression \(9\sqrt{7}\).

This approach is very useful not only in simplifying but also in factoring and multiplying expressions in algebra.

Combining Like Terms

Combining like terms is a central skill in algebra that involves summing terms with the same variable raised to the same power. Terms are considered "like" if all the variables in them are identical. The coefficients of these terms can be added or subtracted as needed.
In the current context, the "terms" are the radical terms \(8\sqrt{7}\) and \(\sqrt{7}\). Both terms share the radical \(\sqrt{7}\), making them like terms.
By identifying these like terms, you can use the distributive property to factor them neatly. This allows you to simplify the expression by gathering the coefficients:

• Add the coefficients of the like terms: \(8 + 1 = 9\).
• Thus, the expression becomes \(9\sqrt{7}\).

Using this method helps streamline more complex expressions throughout algebra and is a vital tool for keeping your work organized and simple. Combining like terms is an essential step in both simplifying and solving equations.