Decimal / Rational Exponents BY HAND

In summary, the conversation is about how to evaluate an arbitrary rational exponent without a calculator. The suggested method is to convert the expression to exponential form and use logarithms and antilogarithms from a mathematical table. Another possible method mentioned is using Newton's method.
  • #1
EebamXela
16
0
I'm reviewing for a test. One of my questions on the review (and incidentally a question I've had in my own mind for a long time) is how do you evaluate an ARBITRARY rational exponent with pencil and paper and no calculator?

The specific problem i was givens is "Solve without a calculator: 3.4[tex]^{2.1}[/tex] "

I know the expression can be broken up to look like this:

3.4[tex]^{2}[/tex] * 3.4[tex]^{0.1}[/tex]

But how in the world can you calculate the tenth root of something on paper.

Or perhaps I'm missing some critical trick or rule or something. Please help.

Can you only practically use Newton's method?
 
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  • #2
Probably this won't help you, but what people in the old days (before calculators) would do, is to convert the expression to
[tex]
3.4^{2.1} = \exp(2.1 \times \log(3.4))
[/tex]
and then look up logarithms and antilogarithms in a table like http://en.wikipedia.org/wiki/Handbook_of_Mathematical_Functions_with_Formulas,_Graphs,_and_Mathematical_Tables" [Broken].
 
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  • #3
This thread may be of some help.
 

1. What is the difference between decimal and rational exponents?

Decimal exponents are numerical values written in decimal form, such as 2.5 or 0.75. Rational exponents are expressed as a fraction, with a base number raised to a fractional power, such as 2^(3/4).

2. How do you convert a decimal exponent to a rational exponent?

To convert a decimal exponent to a rational exponent, you can write the decimal as a fraction with a denominator of 1. For example, 2.5 can be written as 2.5/1. Then, you can rewrite the exponent as a fraction with the same denominator, so 2.5^(3/1) becomes (2.5/1)^(3/1).

3. How do you simplify a rational exponent expression?

To simplify a rational exponent expression, you can use the rules of exponents to rewrite the expression in a different form. For example, if you have (2^(3/4))^2, you can rewrite it as 2^(3/4 * 2) = 2^(3/2).

4. Can you solve decimal or rational exponent problems without a calculator?

Yes, you can solve decimal and rational exponent problems by hand using the rules of exponents and basic arithmetic operations. However, for more complicated problems, a calculator may be helpful.

5. How do you know if you have simplified a rational exponent expression correctly?

You can check if you have simplified a rational exponent expression correctly by plugging in a few values for the base number and exponent and seeing if the simplified expression gives the same result as the original expression. Additionally, you can use a calculator to verify the correctness of the simplified expression.

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