Problem 89 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ 8 \sqrt[5]{7 a^{2}}-7 \sqrt[5]{7 a^{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\sqrt[5]{7 a^{2}}\).

Step 1: Understand the Expression

The expression given is \(8 \sqrt[5]{7 a^{2}} - 7 \sqrt[5]{7 a^{2}}\). This is a subtraction problem involving like terms under the same root.

Step 2: Identify Like Terms

Both terms, \(8 \sqrt[5]{7 a^{2}}\) and \(7 \sqrt[5]{7 a^{2}}\), have the same fifth root base, \(\sqrt[5]{7 a^{2}}\). This means they are like terms and can be combined.

Step 3: Subtract Like Terms

Subtract the coefficients of the like terms while keeping the base and index of the root the same: 8 - 7 = 1, thus the result is \(1 \cdot \sqrt[5]{7 a^{2}}\).

Step 4: Simplify the Coefficient

The coefficient 1 in \(1 \cdot \sqrt[5]{7 a^{2}}\) indicates that you essentially have \(\sqrt[5]{7 a^{2}}\). The expression simplifies to \(\sqrt[5]{7 a^{2}}\).

Key Concepts

Simplifying Radical Expressions

Simplifying radical expressions means breaking them down into more manageable parts. This often involves reducing the expression to its simplest form. When simplifying, it's important to consider the root

• Identify common factors inside the radical, such as perfect squares in square roots.
• Look out for patterns, such as powers of variables, that allow further simplification.

In our example, the expression given is a fifth root, \[\sqrt[5]{7a^{2}}.\]When simplifying expressions like this, you should:

• First check if the numbers or variables inside the radical can be simplified.
• Focus on reducing any coefficients outside the radical, as seen in the problem where the coefficients 8 and 7 were simplified by subtraction.

The key focus is to look for like terms under the same root base, which helps in combining them effectively.

Like Terms

In algebra, "like terms" are terms whose variables and their corresponding exponents are the same. They can be added or subtracted easily. For example, \(8 \sqrt[5]{7a^{2}}\) and \(7 \sqrt[5]{7a^{2}}\) are like terms because:

• The expression under the root is identical: \(\sqrt[5]{7a^{2}}\).
• The root type and its index (which is 5 in this case) are the same.

When working with like terms, focus on the coefficients. This means:

• Subtract or add the coefficients in front of similar radical terms.
• This process simplifies the expression while keeping the root term unchanged.

In practice, combining these terms results in the original problem simplifying to \(\sqrt[5]{7a^{2}}\), indicating efficient simplification due to like terms.

Real Numbers

Real numbers include all the numbers on the number line. This set consists of various types of numbers:

• Rational numbers (like fractions and integers)
• Irrational numbers (like roots and c)

When dealing with radical expressions in math, especially in algebra, the variables are often assumed to be positive real numbers. This assumption helps in:

• Ensuring the expression is defined under all arithmetic operations, including roots and powers.
• Simplifying the process of rationalizing denominators and combining terms.

In the problem, since all variables are positive real numbers, you can freely apply operations like taking roots or powers without encountering undefined or complex numbers. This is crucial for ensuring correctness in simplification steps. The focus on real numbers aids in maintaining clarity and accuracy in algebraic manipulations.