Resolving index laws for fractional exponents

The text may say that ##(-1)^{\frac{5}{3}} = \sqrt[3]{-1} = -1## and so students should understand that ##(-1)^{\frac{10}{6}}## cannot be computed using the same rule because ##(-1)^{\frac{1}{3}}## is not a real number.In summary, when dealing with rational exponents, it is important to be aware of the definition being used and the possible limitations of that definition. In some cases, the definition may not apply to certain calculations, such as when dealing with negative bases.
  • #1
etotheipi
I was just thinking about this earlier and couldn't come up with a good enough resolution. I'm guessing it's a matter of convention more than anything. If we have ##x^{2} = a##, taking the principle root of both sides gives ##\sqrt{x^{2}} = \sqrt{a} \implies |x| = \sqrt{a}##.

Yet evidently if the rule ##{(a^{b})}^{c} = a^{bc}## is taken to be true, then we end up with ##{(x^{2})}^{\frac{1}{2}} = a^{\frac{1}{2}} \implies x = \sqrt{a}##, which disregards the potential negative root.

If we use the definition ##a^{b} = e^{b\ln{a}}##, then negative bases make no sense since the domain of ##\ln{x}## is greater than zero. And, if I write ##a## in complex form, it turns out I can use the normal power rules to get both answers as expected:

##x^{2} = ae^{2n\pi i} \implies x = \sqrt{a} e^{n\pi i} = \pm \sqrt{a}##

Is the failure of this particular index law in the second example then just something we need to be aware of, or is there a resolution of some sort?
 
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  • #2
etotheipi said:
I was just thinking about this earlier and couldn't come up with a good enough resolution. I'm guessing it's a matter of convention more than anything. If we have ##x^{2} = a##, taking the principle root of both sides gives ##\sqrt{x^{2}} = \sqrt{a} \implies |x| = \sqrt{a}##.

Yet evidently if the rule ##{(a^{b})}^{c} = a^{bc}## is taken to be true, then we end up with ##{(x^{2})}^{\frac{1}{2}} = a^{\frac{1}{2}} \implies x = \sqrt{a}##, which disregards the potential negative root.
For rational exponents, the rule above applies only when the base is positive.
For example ##((-4)^2)^{1/2} = 4##, but ##((-4)^{1/2})^2## is not a real number.
etotheipi said:
If we use the definition ##a^{b} = e^{b\ln{a}}##, then negative bases make no sense since the domain of ##\ln{x}## is greater than zero. And, if I write ##a## in complex form, it turns out I can use the normal power rules to get both answers as expected:

##x^{2} = ae^{2n\pi i} \implies x = \sqrt{a} e^{n\pi i} = \pm \sqrt{a}##

Is the failure of this particular index law in the second example then just something we need to be aware of, or is there a resolution of some sort?
 
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  • #3
etotheipi said:
I was just thinking about this earlier and couldn't come up with a good enough resolution. I'm guessing it's a matter of convention more than anything.

An underlying problem is that, in texts treating only the real numbers, ##a^r## is not defined for rational numbers ##r##. Instead, ##a^r## is defined in a way that depends on how the rational number ##r## is denoted.

For example, consider ##r = 5/3 = 10/6 ##. Typical secondary school texts in the USA give the definition:

For integers ##m,n## if ##a^{\frac{1}{n}}## is a real number then ##a^{\frac{m}{n}}## is defined to be ##(a^{\frac{1}{n}})^m##.

Using this definition, students may be expected to compute ##(-1)^{\frac{5}{3}}## but they are expected to say that ##(-1)^{\frac{10}{6}}## does not exist - or perhaps the text avoids confusing the students by never asking for such a computation.
 
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1. What are fractional exponents?

Fractional exponents are a way of representing powers or roots of numbers that are not whole numbers. They are written in the form of a fraction, where the numerator is the power and the denominator is the root.

2. How do I resolve index laws for fractional exponents?

To resolve index laws for fractional exponents, you can use the following rules:

  • Rule 1: When multiplying powers with the same base, add the exponents.
  • Rule 2: When dividing powers with the same base, subtract the exponents.
  • Rule 3: When raising a power to another power, multiply the exponents.
  • Rule 4: When taking the root of a power, divide the exponent by the root.

3. Can fractional exponents be negative?

Yes, fractional exponents can be negative. This indicates that the number is being raised to a negative power, which is equivalent to taking the reciprocal of the number to the positive power.

4. How do I simplify expressions with fractional exponents?

To simplify expressions with fractional exponents, you can use the rules mentioned above and also simplify any fractions in the exponents. Additionally, you can convert fractional exponents to radical form to make the simplification process easier.

5. What is the difference between fractional exponents and radical expressions?

Fractional exponents and radical expressions are two different ways of representing the same concept of powers and roots. Fractional exponents are written in the form of a fraction, while radical expressions use the radical symbol (√) to represent the root. However, they can be converted to each other using the rules mentioned above.

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