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Chapter 10: Problem 48

Write an equivalent expression using exponential notation. $$ \frac{3 a}{\sqrt[5]{c^{2}}} $$

Short Answer

Expert verified
\( 3ac^{-\frac{2}{5}} \)

Step by step solution

01

Understand the given expression

The given expression is \( \frac{3a}{\root{5}{c^2}} \). The goal is to rewrite this using exponential notation.
02

Rewrite the radical as an exponent

Recall that a root can be written as a fractional exponent. Therefore, \( \root{5}{c^2} \) can be rewritten as \( c^{\frac{2}{5}} \) in exponential notation.
03

Substitute the fractional exponent

Replace \( \root{5}{c^2} \) in the original expression with \( c^{\frac{2}{5}} \). This gives: \( \frac{3a}{c^{\frac{2}{5}}} \).
04

Simplify the expression using negative exponents

Rewrite the denominator with a negative exponent. This gives: \( 3a \times c^{-\frac{2}{5}} \).
05

Combine the terms

Write the expression in its final simplified form: \( 3ac^{-\frac{2}{5}} \).

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Key Concepts

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

fractional exponents
When we're dealing with exponents, sometimes they aren't just whole numbers. Fractional exponents can represent roots, which is handy for simplifying expressions. Consider the fraction exponent as a signpost: the denominator tells you the root, and the numerator tells you the power. This means \(c^{\frac{2}{5}}\) can be seen as the 5th root of \(c\) squared (or \(\sqrt[5]{c^2}\)). Remembering this transformation is key: \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\). This makes it simpler when you're switching from a radical to an exponent form. Plus, it streamlines the math!
negative exponents
Negative exponents can seem tricky at first, but they're just another tool to help simplify expressions. A negative exponent like \(c^{-\frac{2}{5}}\) essentially means the reciprocal of the positive exponent. So, \(c^{-\frac{2}{5}} = \frac{1}{c^{\frac{2}{5}}}\).
This is important when you need to write expressions without fractions. By converting to negative exponents, we can express \(\frac{3a}{c^{\frac{2}{5}}}\) as \(3a \times c^{-\frac{2}{5}}\). This method keeps your expression in a neat exponential form, making it easier to handle in further calculations.
simplifying expressions
Simplifying expressions is all about making them more manageable and easier to work with. With fractional and negative exponents, the process becomes a straightforward transformation. Starting from \(\frac{3a}{\sqrt[5]{c^2}}\), by using the rules of fractional exponents, you rewrite the denominator. Then, use negative exponents to eliminate the fraction: \(\frac{3a}{c^{\frac{2}{5}}} = 3a \times c^{-\frac{2}{5}}\).
Breaking it down this way clarifies each step and leaves you with a simplified exponential expression: \(3ac^{-\frac{2}{5}}\). This form is not only simpler but easier to use in further mathematical operations.

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