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How would you define what a rational exponent is is there such a thing as an irrational exponent exp

What does it really mean?

So it turns out that this is actually the exact same formula as the method given above. The main advantage of this re-expression of this method is that it gives us the opportunity to simplify step 3.

For example, if the division in step 3 is only accurate to, say, one digit or bit, then the algorithm will converge exactly one bit or digit per step. So depending on how expensive the division is and how much better we might be able to do with a low accuracy divider, this approach can be better than the first one.

So a simple question is, can we remove the divisions altogether?

You already know of one relationship between exponents and radicals: the appropriate radical will "undo" an exponent, and the right power will "undo" a root. For example: But there is another relationship — which, by the way, can make computations like those above much simpler. The way Rudin goes about it (if you care about that sort of thing), is by limits of rational exponents. So if a is positive, a b = sup{a x where x. When serious work needs to be done with radicals, they are usually changed to a name that uses exponents, so that the exponent laws can be used.

Ironically, we can, not by estimating the square root, but the reciprocal of the square root inverse square root. So let us start with an estimate xest of the reciprocal square root and iterate it through the following formula: Unfortunately, we have somewhat tighter requirements on the initial estimate for this to converge successfully.

Although it may require more iterations, the benefit of removing the use of any divisions easily compensates for this. Well here, fortunately, we can exploit the IEEE format for floating point numbers. Numbers other than 0, inf, -inf, NaN and denormals are represented as: So we can assume sign is 1, and the square root is basically: So we need only check this estimate for, say, the range 1 to 4.

Between the range 1 and 2-epsilon, the worst deviation is at 2-epsilon which delivers an estimate of 0. This function is periodic with respect to the exponent but breaks into two distinct cases depending on whether the exponent is even or odd.

So lets look at each case one by one. But we will use the same formulaic form as above with a different constant: So this motivates us to consider a formula of the form: In fact let us examine the exact encoding of the IEEE format: Thus, a right shift of these bits will: Preserve the sign as 0 which means positive in IEEE Divide the biased exponent by 2 and add a constant to it Divide the mantissa by 2 but also add 0.

Negating these bits which is essentially complementing them will essentially have the effect of negating the mantissa and the exponent. Looking at the reformulated reciprocal square root formula we see that this is exactly what we are looking for. Thus we should be able to find a constant J such that: This seems to be a stunning simplification, however by ordering the bits as shown the IEEE specification included consideration into its very design for being able to perform exactly this trick.

Solving for J is straight forward and breaks down into the two major cases for 32 bit and 64 bit numbers. The 32 bit floating point representation of 4 is 0x you can use the IEEE format conversion tool here to see thisand the representation of 0.

This solves to J being 0xBE So we can see the rough range for values of J. An example of such a scanner can be downloaded here. The results are accurate to 4 bits, which means at most 4 iterations are required for 32 bit floating point and 5 for 64 bit floating point.

Its time for a graph: The red line is our estimator. The horizontal range is from 1 to 4. This is where a carry happens into the exponent. We can see that at least visually, this approximation is fairly close. An explanation of this method and how an implementation based on it made it into the Quake source code.You already know of one relationship between exponents and radicals: the appropriate radical will "undo" an exponent, and the right power will "undo" a root.

For example: But there is another relationship — which, by the way, can make computations like those above much simpler.

Formula when the fraction in the exponent is not 1 Below is a specific example illustrating the formula for fraction exponents when the numerator is not one. There are two ways to simplify a fraction exponent such 2/3.

The Power Rule for Irrational Exponents There is a real problem when it comes to considering power functions with irrational exponents. If m and n are postive integers, then the meaning of x m/n is fairly clear: take the .

Imaginary numbers always confused me. Like understanding e, most explanations fell into one of two categories. It’s a mathematical abstraction, and the equations work out.

Deal with it. It’s used in advanced physics, trust us. Felicia Young: Week 3 Discussion Question #1 • How would you define what a rational exponent is?

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Is there such a thing as an irrational exponent? Explain. • What are the two steps for simplifying radicals? Can either step be deleted?

If you could add a step that might make simplifying radicals easier or easier to understand, what step. If fractions get you down you may want to go to Beginning Algebra Tutorial 3: Fractions. To review exponents, you can go to Tutorial Exponents and Scientific Notation Part I and Tutorial Exponents and Scientific Notation Part II.

Let's move onto rational exponents and roots.. After completing this tutorial, you should be able to. Rewrite a .

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