Problem 96 - College Algebra Essentials

Question

In Exercises \(91-100,\) simplify using properties of exponents. $$\left(x^{\frac{4}{5}}\right)^{5}$$

Step-by-Step Solution

Answer

\(x^4\).

Step 1: Identify the exponent rule

Identify the power-to-power rule in exponent properties. The power-to-power rule states that \((a^m)^n=a^{m*n}\). Let's apply this to our problem.

Step 2: Apply rule to problem

According to the power-to-power rule, multiply the exponents. In our case, the exercise simplifies to \(x^{(4/5)*5}\).

Step 3: Simplify Exponent

Now, multiply the exponents. \(4/5 * 5 = 4\) in our case, hence \(x^{(4/5)*5}\) simplifies to \(x^4\).

Step 4: Final Answer

The final simplification of the expression \((x^{4/5})^{5}\) is \(x^4\).

Key Concepts

Simplifying Exponential Expressions

When faced with exponential expressions, the goal is to make them as straightforward as possible. Simplifying such expressions often involves applying various exponent rules to reduce the complexity of the expression.

One common example, much like the textbook exercise \(\left(x^{\frac{4}{5}}\right)^{5}\), requires us to use our knowledge of exponent rules unwaveringly. To simplify an expression like this, you analyze the base and the exponent, and strategically apply the rules to arrive at a more elementary form.

For better understanding, always remember the base of an exponential expression is the number or variable that is being raised to a power, which is the exponent. It's crucial to pay attention to the exponent's form, as it could be an integer, a fraction, or even another expression in itself.

With a keen eye and confident application of the appropriate rules, such as the power-to-power rule, one can drastically streamline exponential equations, making them easier to interpret and, ultimately, solve.

Power-to-Power Rule

In an exponential expression, the power-to-power rule is an essential mechanism for simplification. When an exponent is raised to another exponent, like \(a^{m}\)^{n}\, you multiply the two exponents together. It’s critical to keep the base the same while performing this operation.

This rule highlights the efficiency of exponents in encapsulating repeated multiplications. In the provided exercise example, \(\left(x^{\frac{4}{5}}\right)^{5}\), we apply this rule by multiplying the exponents \(\frac{4}{5}\) and 5, which conveniently results in a whole number, simplifying the expression significantly.

Remember that this rule applies across all types of bases—numerical, algebraic, and even those with multiple variables—making it a versatile tool in your exponential expressions toolkit.

Exponent Multiplication

Multiplication of exponents comes into play when the base remains constant, and you encounter multiple exponential terms. In essence, when multiplying, if the bases are alike, you can add the exponents, depicted as \(a^{m} * a^{n} = a^{m+n}\). However, this should not be confused with the power-to-power rule.

In the original exercise \(\left(x^{\frac{4}{5}}\right)^{5}\), we're not directly adding exponents but instead multiplying them because of the structure of the expression—a power raised to another power. It's this subtlety between multiplication and addition of exponents which often confuses students.

By holding a firm grasp on these rules, solving exponential expressions can become a series of simple, orderly steps rather than a perplexing ordeal. Apply these rules consistently, and advanced algebra no longer seems quite as daunting.