Problem 89 - 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 \(\sqrt[5]{7a^2}\).

Step 1: Understand the Expression

The expression given is: \(8 \sqrt[5]{7a^2} - 7 \sqrt[5]{7a^2}\). This means we have two terms with similar structures and we are asked to simplify by combining them if possible.

Step 2: Identify Like Terms

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

Step 3: Combine Like Terms

Since the terms are like terms, we can combine their coefficients. We calculate: \(8 - 7 = 1\). The expression simplifies to \(1 \times \sqrt[5]{7a^2}\), which can be written as \(\sqrt[5]{7a^2}\).

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that include a root symbol, such as a square root or in this case, a fifth root. These symbols indicate that a number or expression is being raised to a fractional power. Generally, a radical expression is composed of two parts: the radicand, which is the number or expression inside the radical; and the index, which is the small number outside the root that indicates the degree of the root.

• The expression \(\sqrt[5]{7a^2}\) is a fifth root radical expression, where \(7a^2\) is the radicand and 5 is the index of the radical.
• Radicals can often be simplified if the radicand can be expressed as a power of the index.
• Simplifying radical expressions often involves combining like terms when possible to reduce the expression to its simplest form.

Understanding radical expressions is crucial when working with problems involving roots, as it helps in simplifying and solving complex algebraic problems.

Like Terms

In algebra, like terms refer to terms that contain the exact same variables raised to the same powers. Identifying like terms in an expression is essential for simplifying because they can be combined.

• In the expression \(8\sqrt[5]{7a^2} - 7\sqrt[5]{7a^2}\), both terms contain the radical \(\sqrt[5]{7a^2}\), making them like terms.
• Since they are like terms, we can directly combine them by adding or subtracting their coefficients.
• Combining like terms helps to simplify expressions by reducing the number of terms.

Remember, only terms with the exact same variables and exponents are considered like terms, which allows for efficient simplification.

Coefficients

The coefficient in a term is the numerical factor that is multiplied by the variable portion of the term. Recognizing coefficients in algebra is an important skill, especially in simplifying and combining expressions.

• For instance, in \(8\sqrt[5]{7a^2}\), the coefficient is 8, while in \(-7\sqrt[5]{7a^2}\), it is \(-7\).
• When simplifying an expression with like terms, you simply add or subtract the coefficients and bring down the constant part, which in this case, is a radical.
• In our problem, the calculation of \(8 - 7 = 1\) represents combining the coefficients to simplify the expression.

By mastering coefficients, you gain the ability to efficiently manipulate and simplify algebraic expressions.

Fifth Roots

Fifth roots are a type of radical where the radicand is raised to the power of one-fifth. This is denoted by the index 5 in the root symbol. Understanding how to handle fifth roots is essential, especially when simplifying expressions that involve exponents and radical components.

• The fifth root of a number \(x\) is represented as \(\sqrt[5]{x}\).
• For example, \(\sqrt[5]{32} = 2\), because \(2^5 = 32\).
• In the expression \(\sqrt[5]{7a^2}\), the fifth root is applied to the entire radicand \(7a^2\).

Fifth roots are part of a wider family of roots used in algebraic expressions, such as square roots or cube roots, and serve the purpose of simplifying expressions where higher indices are involved.