Problem 34 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$\left(x^{-6}\right)^{4}$$

Step-by-Step Solution

Answer

The simplified version of \( \left( x^{-6} \right) ^{4} \) is \( 1/x^{24} \).

Step 1: Identify base variable and its power

Here, it can be identified that base variable is 'x' and it is raised to the power of -6, followed by a higher power of 4. \(x^{-6}\) is raised to the power of 4.

Step 2: Apply the power of a power rule

According to the power-of-a-power rule, powers gets multiplied when they are stacked up. Here, -6 gets multipled by 4 producing -24. Thus the expression would transform into \(x^{-24}\)

Step 3: Interpret the negative in the exponent

Here, the negative exponent would mean that it is in the denominator of a fraction. Thus it clears the negative exponent by flipping the base to the denominator. Thus it can be written as \(1/x^{24}\)

Key Concepts

Power of a Power Rule

One of the fundamental rules of working with exponents is the **power of a power rule**. This rule comes in handy when you are dealing with expressions where an exponent is raised to another exponent. For instance, if you come across an expression like \((a^m)^n\), the power of a power rule tells us that the exponents should be multiplied together. In other words, it becomes \(a^{m \cdot n}\). This rule simplifies expressions by condensing a multi-layered power structure into a single exponent.

When you encounter something like \( (x^{-6})^4 \), you apply this rule by multiplying -6 and 4, resulting in \(x^{-24}\). This multiplication of exponents makes it easier to work with by reducing potentially cumbersome and nested expressions into a manageable single-layer expression.

Negative Exponents

Negative exponents can initially seem a bit confusing, but with a little practice, they become quite straightforward. A negative exponent indicates that the base is on the "wrong side" of a fraction line. More precisely, if you see a negative exponent, it means you should take the reciprocal of the base.

For example, if you have \(x^{-24}\), it is equivalent to \(\frac{1}{x^{24}}\). Negative exponents are a powerful tool in mathematics, as they allow you to express very small numbers conveniently without moving between the numerator and the denominator. Understanding this will help you manipulate expressions effectively, especially when combined with other exponent rules.

Simplifying Expressions

Simplifying mathematical expressions is all about making them easier to understand and work with. With exponents, simplification often involves applying rules such as the power of a power rule and the handling of negative exponents.

Consider the expression \((x^{-6})^4\). Initially, it may look complex, but the simplification process breaks it down into straightforward steps:

• Apply the power of a power rule to get \(x^{-24}\).
• Convert the negative exponent by moving the base to the denominator, resulting in \(\frac{1}{x^{24}}\).

Simplifying expressions can aid in better problem-solving and clearer communication of mathematical ideas. By stripping down complex expressions, we allow both ourselves and others to understand the underlying concepts more swiftly. This approach can be particularly beneficial when tackling higher-level math problems.