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Chapter 0: Problem 125

What does \(a^{\frac{m}{n}}\) mean?

Short Answer

Expert verified
The expression \(a^{\frac{m}{n}}\) means 'a' raised to the power 'm' and then taking the nth root of the result. Alternatively, it denotes the nth root of 'a' raised to the power 'm'. It combines both the operations of exponentiation and root finding.

Step by step solution

01

Recognise the combination of exponent and root

Firstly, realise that \(a^{\frac{m}{n}}\) combines two operations: exponentiation and finding a root. The numerator 'm' is the exponent, whereas the denominator 'n' is the root.
02

Express it in exponential form

Express the number 'a' to the power of 'm' which can be written as \(a^{m}\).
03

Express it in root form

Find the nth root of \(a^m\). This is equivalent to taking the whole result of the expression \(a^m\), and then calculating its nth root. In other words, it can also be expressed as \(\sqrt[n]{a^m}\).

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Most popular questions from this chapter

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$\text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. }$$ How old is the son? b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}+2^{-1}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?

Evaluate each expression. $$\sqrt[3]{\sqrt[4]{16}+\sqrt{625}}$$

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Simplify by reducing the index of the radical. $$\sqrt[12]{x^{4} y^{8}}$$

a. Find \(\sqrt{16} \cdot \sqrt{4}\) b. Find \(\sqrt{16 \cdot 4}\) c. Based on your answers to parts (a) and (b), what can you conclude?
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