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Chapter 7: Problem 32

Evaluate each exponential. $$ (-243)^{2 / 5} $$

Short Answer

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Step by step solution

01

- Understand the Exponent

The exponent \(\frac{2}{5}\) can be split into two parts: 2 (numerator) and 5 (denominator). The denominator 5 represents the root, and the numerator 2 represents the power.
02

- Rewrite the Expression

Rewrite \((-243)^{2/5}\) using roots and powers: \((-243)^{2/5} = ( (-243)^{1/5} )^2\). This means we first find the 5th root of -243 and then square it.
03

- Calculate the 5th Root of -243

The 5th root of -243 is -3, because \((-3)^5 = -243\). Thus, \((-243)^{1/5} = -3\).
04

- Square the Result

Now, we need to square the result of the previous step: \((-3)^2 = 9\).

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Key Concepts

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

fractional exponents
Fractional exponents can be a bit tricky at first, but they are quite manageable once you split them into parts. A fractional exponent like \(\frac{2}{5}\) consists of a numerator and a denominator. The numerator (here, 2) indicates the power, and the denominator (here, 5) indicates the root. This means \((-243)^{2/5}\) can be rewritten as \(((-243)^{1/5})^2\). When dealing with fractional exponents, always remember to take the root first and then apply the power. This method makes the task less daunting and more straightforward.
radicals and roots
Radicals and roots are fundamental in understanding expressions like \((-243)^{2/5}\). To clarify, radicals and roots are essentially the same. For instance, the 5th root of -243 is represented as \(\root5\backslash!-243\) or \((-243)^{1/5}\). In our case, the 5th root of -243 gives us -3 because \((-3)^5 = -243\). This calculation helps us convert roots to fractions, allowing for easier computations. Understanding this relationship is essential as it transforms complex expressions into simpler forms that are easier to manage.
exponential rules
Using exponential rules simplifies calculations and boosts your understanding. One key rule to remember is that \((a^m)^n = a^{mn}\). This means when we have \((-243)^{2/5}\), it turns into \(((-243)^{1/5})^2\). Let's break it down: First, find \((-243)^{1/5}\), which is -3, and then raise it to the power of 2. So, \((-3)^2 = 9\). By following these rules, exponential expressions become more approachable and less intimidating. Don't forget to apply the rules step by step for accurate results.

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Most popular questions from this chapter

Graph each circle. Identify the center and the radius. \(x^{2}+y^{2}=16\)

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt[6]{y^{5}} \cdot \sqrt[3]{y^{2}} $$

Simplify each expression. Assume that all variables represent positive real numbers. $$ -8 y^{11 / 7}\left(y^{3 / 7}-y^{-4 / 7}\right) $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{r-9}{\sqrt{r}-3} $$

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{256 z^{12}}\)
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