Problem 490 - Elementary Algebra

Question

In the following exercises, simplify. \(\sqrt[7]{8 p}+\sqrt[7]{8 p}\) \(3 \sqrt[3]{25}-\sqrt[3]{25}\)

Step-by-Step Solution

Answer

The simplified forms are \(2\sqrt[7]{8 p}\) and \(2\sqrt[3]{25}\).

Step 1: Simplify the First Expression

Combine like terms for the expression \( \sqrt[7]{8 p} + \sqrt[7]{8 p}\). Since both terms are the same, you simply add their coefficients: \(1\sqrt[7]{8 p} + 1\sqrt[7]{8 p} = 2\sqrt[7]{8 p}\).

Step 2: Simplify the Second Expression

Combine like terms for the expression \(3\sqrt[3]{25} - \sqrt[3]{25}\). Subtract the coefficients: \(3 \sqrt[3]{25} - 1\sqrt[3]{25} = 2\sqrt[3]{25}\).

Key Concepts

Understanding Radicals

A radical expression contains a root symbol, such as a square root or cube root. In mathematics, radicals are a way of expressing roots of numbers. For example, \(\root{3}{8}\) represents the cube root of 8. Radicals help to find the number, which when multiplied by itself a certain number of times, equals the original number. In the provided exercise, we work with the seventh root \(\root{7}{8p}\) and the cube root \(\root{3}{25}\). Remember, different roots do not combine together; only like radicals can be simplified or added together.

• Example: The cube root of 8 is 2, since \(2 \times 2 \times 2 = 8\).
• Example: The seventh root of 128 is approximately 2, since \(2^7 = 128\).

Working with radicals requires recognizing these root patterns and ensuring only similar radicals are combined.

Identifying Like Terms

Like terms are terms that contain the same variables raised to the same power. In simpler terms, they are terms that can be combined through addition or subtraction. When simplifying expressions, combining like terms reduces the expression to its simplest form. In the given exercise, \(\root{7}{8p}\) and \(\root{7}{8p}\) are like terms because they both contain the same radical. Similarly \(\root{3}{25}\) can be combined with \(\root{3}{25}\) because they are both cube roots of 25.

• Example: \(3x + 2x\) are like terms because they both involve \(x\).
• Example: \(\root{5}{32} + \root{5}{32}\) can be combined as they are like terms.

By adding or subtracting like terms, you bring simplicity and clarity to your expressions.

Understanding Coefficients

In algebra, a coefficient is the numerical factor in a term that includes a variable or radical. It is the number multiplying the variable or radical expression. For instance, in the term \(3\root{3}{25}\), the coefficient is 3.
In the problem, when we combined \(\root{7}{8p} + \root{7}{8p}\), we noted that both terms have an understood coefficient of 1, giving us \(2 \root{7}{8p}\). Similarly, for \((3 \root{3}{25} - \root{3}{25})\), the coefficients are 3 and 1, respectively, which when subtracted give us 2. The coefficients are subtracted or added first before simplifying the resulting term.

• Example: In \(5x\), 5 is the coefficient.
• Example: In \(2 \root{4}{7}\), 2 is the coefficient.

Understanding coefficients makes it easier to simplify and combine similar terms effectively.