Question

(01.02 HC)
Write each expression with a single rational exponent. Show each step of your process. Which expressions are equivalent? Justify your reasoning.

Answer 1

By applying radical, power and rational properties we have the following results after simplifying expressions:

1. $x^{\frac{3}{4} }$
2. x
3. $x^{\frac{11}{10} }$
4. x

How to reduce radical and rational expressions into a single power expression with rational exponent

In this question we need to simplify four given expressions by using the following radical and power properties:

1. $\sqrt[n]{x} = x^{\frac{1}{n} }$     (1)
2. $x^{m+n} = x^{m}\cdot x^{n}$     (2)
3. $x^{m-n} = \frac{x^{m}}{x^{n}}$     (3)
4. $(x^{m})^{n} = x^{m\cdot n}$     (4)

Now we proceed to simplify each of the four expressions:

1. $\sqrt[4]{x^{3}} = x^{3/4}$
2. $\frac{1}{x^{-1}} = (x^{-1})^{-1} = x$
3. $\sqrt[10]{x^{5}\cdot x^{4}\cdot x^{2}} = \sqrt[10]{x^{11}} = x^{\frac{11}{10} }$
4. $x^{\frac{1}{3} }\cdot x^{\frac{1}{3} }\cdot x^{\frac{1}{3} } = x$

To learn more on power functions: https://brainly.com/question/18719083

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