Problem 592 - Elementary Algebra

Question

In the following exercises, simplify. $$ \left(a^{24}\right)^{\frac{1}{6}} $$

Step-by-Step Solution

Answer

The simplified result is \(a^4\).

Step 1: Understand the Problem

The problem requires simplification of the exponential expression \(\left(a^{24}\right)^{\frac{1}{6}}\). This involves applying the rules of exponents.

Step 2: Apply the Power of a Power Rule

Use the power of a power rule, which states \((x^m)^n = x^{m \cdot n}\). For our expression, identify \(m = 24\) and \(n = \frac{1}{6}\).

Step 3: Multiply the Exponents

Multiply the exponents: \(24 \cdot \frac{1}{6} = 4\). So, the expression simplifies to \(a^4\).

Step 4: Write the Final Answer

The simplified form of \(\left(a^{24}\right)^{\frac{1}{6}}\) is \(a^4\).

Key Concepts

power of a power rule

When working with exponents, the power of a power rule is extremely useful. This rule states that if you have an expression where a power is raised to another power, like \( (x^m)^n \), you simply multiply the exponents together. For example, in \( (a^{24})^{\frac{1}{6}} \), you need to identify the exponents. Here, \( m = 24 \) and \( n = \frac{1}{6} \). According to the rule, you multiply these: \( 24 \times \frac{1}{6} = 4 \). This gives you the simplified exponent: \( a^4 \).

This rule makes complex exponentiation easier and helps in breaking down difficult problems into manageable steps.

exponentiation

Exponentiation is a fundamental operation in algebra involving numbers called 'exponents.' An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \( a^3 \), 'a' is the base, and '3' is the exponent, which means \( a^3 = a \times a \times a \).

Understanding how to manipulate and simplify expressions with exponents is crucial. For example, in the problem \( (a^{24})^{\frac{1}{6}} \), exponentiation rules help you simplify this down to \( a^4 \).

Without these rules, working through algebraic expressions would be more complex and time-consuming.

simplifying expressions

Simplifying expressions means rewriting them in a simpler or more convenient form without changing their value. Key techniques include factoring, combining like terms, and using the rules of exponents. For instance, in the given problem, the expression \( (a^{24})^{\frac{1}{6}} \) can be simplified by applying the power of a power rule.

This rule reduces the expression \( (a^{24})^{\frac{1}{6}} \) to \( a^4 \), making it much easier to handle. Simplifying expressions is an essential skill in algebra and helps to solve equations, evaluate functions, and find solutions efficiently.

algebra basics

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It forms the foundation for more advanced studies in math and science. Basic algebraic concepts include understanding variables, constants, expressions, and equations.

The expression \( (a^{24})^{\frac{1}{6}} \) is an excellent example of how basic algebraic rules come into play. Applying the power of a power rule simplifies it to \( a^4 \). Learning these basics allows you to explore more complex problems and develop problem-solving skills. Being comfortable with the basics of algebra paves the way for success in more advanced topics.