The worksheets can be made in html or PDF format both are easy to print. Options include negative and zero exponents, and using fractions, decimals, or negative numbers as bases. These worksheets are most useful in 6th, 7th, and 8th grade, when exponents are introduced and practiced.

Binary Fractional Numbers Fractions present another number representation problem. How can one represent fractional quantities using bits? Representing fractions can be solved in the same way that positive powers of 2 represented integers, use negative powers of two to added up to approximate fractional quantities.

The small negative powers of two are: Thus you can pick some part of a bit computer word and decide where the binary point should be.

This gives 1 bit of sign, 15 bits of integer, and 16 bits of fraction. If you use this technique you must convert numbers from the bit representation to decimal representation correctly. Pascal and C do not provide this flexibility in conversion, but some other programming languages do.

However one can always write conversion subroutines in other languages that do work properly so that you can both input fractional quantities and output them properly. Note that the rules for binary arithmetic on these fractional numbers are the same as it write answer in positive exponents for integers, so it requires no change in the hardware to deal with binary fractions.

That is this fractional representation is isomorphic to the binary twos-complement integer representation that the machine uses. It only requires that you think the numbers are fractions and that you understand and treat properly the binary-point the binary equivalent of the decimal point.

For example you can only meaningfully add two numbers if the binary point is aligned. Of course, the computer will add them no matter where you think the binary point is.

This is just like decimal arithmetic where you must add two numbers with the decimal points aligned if you want the right answer. One can shift the binary point of a number by multiplying or dividing by the proper power of two, just as one shifts the decimal point by multiplying or dividing by a power of ten.

Most machines also provide arithmetic shift operations that shift the bit representation of the number right or left more quickly than a multiply instruction would. An arithmetic right shift of 3 is a binary division by 23 or 8; a left shift is a binary multiplication. Arithmetic right shifts usually copy the sign bit so that negative numbers stay negative.

Arithmetic left shifts introduce zeros from the right side of the number. If the sign bit changes in an arithmetic left shift then the number has overflowed. The binary point can be represented by counting the number of binary fractional digits in the number.

Multiplication of a number with binary point of 5 and one with a binary point of 3 will give a number with a binary point of 8.

That is binary numbers can be represented in general as having p binary digits and q fractional digits. The result of binary arithmetic for A with and B with gives the worst case p: One reason that this scaled integer arithmetic is not popular is that unless the program scales numbers using multiplication and division by powers of the radix to make sure that the two operands of addition and subtraction have the same q, answers will be wrong.

Note you can do this in Pascal or C because it only involves the way you think about the the numbers you are using. One is not restricted to scaling in the binary representation, one can also scale in the decimal representation. The rules for decimal scaling are the same as in the table above.

You treat it that way in the machine, multiply it by 2.

Again you must be careful to only add and subtract numbers with the same decimal scale, and carefully scale the numbers around multiplication and division to preserve both the high-order digits while maintaining the required precision after the decimal point.

Fractional Quantities as Rational Numbers. One can represent any number as the ratio of two integers. To do this with bits, rational fractional quantities are represented as two integers, one representing the numerator and other the denominator of a fraction that equals the number.

Multiplication and division of these numbers is easy. Addition and subtraction requires finding the common denominator the denominator of the result and then adding or subtracting the adjusted numerator. The major problem with this representation of numbers is that it requires twice as much space, and comparison is costly because it requires two divisions.

Non-rational numbers must be approximated by a rational. Rationals whose denominator gets too large for the integer size available must also be approximated. Error analysis is non-trivial for algorithms represented this way. Few computers use it as a hardware method of representing numbers, but the algorithms for doing rational arithmetic are not difficult to code in any language using standard binary integers, and structures.

Floating Point Numbers For computer languages that do not do fixed scaling of integers, like Pascal and C, the easiest way of handling fractions is with floating point numbers, sometimes called real numbers. Using this technique a number is represented in bits by three parts: This is similar to scientific notation used to represent large or small numbers e.

The sign is negative, the exponent is 8 and the fraction is 0.Purplemath. Once you've learned about negative numbers, you can also learn about negative powers.A negative exponent just means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side.

To make a negative exponent positive, move the base and its exponent to the opposite location in the fraction (reciprocation).

That is, a-n = 1/a n and 1/a-n = a n. Using the laws of exponents, we can prove the above by the following: a n • a-n = a n+(-n) = a 0 = 1 ; therefore, a-n = 1/a n (where a≠0).

With this, (2xy 3)-2 = 1/(2xy 3) 2. Dec 15, · Best Answer: Easy, there is an exponent law that says x^-n = 1/x^n So for positive exponents like x^n then it becomes 1/x^-n 3^-5 = 1/3^5 and 1/3^5 * 3 is how you would show this with the 3 in the problem So just think of putting a ONE over the expression, changing the negative to a positive exponent.

This Status: Resolved. A positive exponent shows that the decimal point is shifted that number of places to the right. A negative exponent shows that the decimal point is shifted that number of places to the left.

In scientific notation, the digit term indicates the . Exponents are an essential part of basic math and appear on almost every high school exam and college entrance exam. Tips and Shortcuts Common Exponents Tests with Exponents Practice Questions Answer Key Tutorials.

Audio Version of this Post. Houston Community College TSI Pre-Assessment Activity TSI Home; Optional Resources we need to create a fraction and put the exponential expression in the denominator and make the exponent positive.

For example, Simplify each of the following expressions using the zero exponent rule for exponents. Write each expression using only.

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Transformations, Inverses, Compositions, and Inequalities of Exponents/Logs – She Loves Math