Problem 27 Evaluate each exponential. $$ ... [FREE SOLUTION]

Chapter 10: Problem 27

Evaluate each exponential. $$ 81^{3 / 4} $$

Short Answer

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Step by step solution

Understand the exponent

The exponent can be separated into a fraction, where the numerator represents a power and the denominator represents a root. Here, the given expression is $ 81^{3/4} $. The numerator is 3, and the denominator is 4, which means we need to find the fourth root of 81 and then raise the result to the power of 3.

Find the fourth root

Calculate the fourth root of 81. The fourth root of a number is a value that, when raised to the power of 4, gives the original number. Since $ 3^4 = 81 $, the fourth root of 81 is 3.

Raise the root to the power

Now, take the fourth root result and raise it to the power of 3. Hence, $ 3^3 = 27 $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fractional Exponents

Fractional exponents might seem tricky at first, but they are very useful for expressing roots and powers in a single step. The expression of the form $a^{m/n}$ means that we first take the nth root of a and then raise it to the mth power. For instance, $81^{3/4}$ breaks down into the 4th root of 81 raised to the power of 3. Understanding this will help you simplify complex exponent expressions. This fractional notation is a neat way to handle both roots and powers in one go.

Fourth Root

The fourth root of a number is a value that gives the original number when it is raised to the power of 4. For example, to find the fourth root of 81, consider which number raised to the power of 4 equals 81. Here, $3^4 = 81$, so the fourth root of 81 is 3. In general, finding the fourth root means solving for $x$ in the equation $x^4 = a$ where $a$ is the original number. Identifying roots simplifies the mathematical process, making it easier to handle higher powers later.

Raising to a Power

After identifying the base using the root, the next step is to raise it to the specified power indicated by the fractional exponent. In the example of $81^{3/4}$, after finding that the fourth root of 81 is 3, we then raise 3 to the power of 3, which is 27. Mathematically, this is written $3^3 = 27$ and is calculated by multiplying 3 by itself two more times: 3 * 3 * 3. Mastering the operation of raising numbers to powers is essential in understanding and solving exponential expressions effectively.

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91影视

Evaluate each exponential. $$ 81^{3 / 4} $$

Short Answer

Step by step solution

Understand the exponent

Find the fourth root

Raise the root to the power

Key Concepts

Fractional Exponents

Fourth Root

Raising to a Power

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