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Powers, Exponents, and Roots

Powers of Negative Numbers, Decimals, and Fractions

Problems

Problems

Powers of Negative Numbers

Since an exponent on a number indicates multiplication by that same number, an exponent on a negative number is simply the negative number multiplied by itself a certain number of times:

(- 4)3 = - 4× -4× - 4 = - 64
(- 4)3 = - 64 is negative because there are 3 negative signs--see Multiplying Negatives.

(- 5)2 = - 5× - 5 = 25
(- 5)2 = 25 is positive because there are 2 negative signs.

Since an odd number of negative numbers multiplied together is always a negative number and an even number of negative numbers multiplied together is always a positive number, a negative number with an odd exponent will always be negative and a negative number with an even exponent will always be positive. So, to take a power of a negative number, take the power of the (positive) opposite of the number, and add a negative sign if the exponent is odd.


Example 1: (- 3)4 = ?

1. Take the power of the positive opposite. 34 = 81 .
2. The exponent (4) is even, so (- 3)4 = 81 .

Example 2: (- 7)3 = ?

1. Take the power of the positive opposite. 73 = 343
2. The exponent (3) is odd, so (- 7)3 = - 343 .

Powers of Decimals

When we square 0.46, we must remember that we are multiplying 0.46×0.46 , not 0.46×46 . In other words, the result has 4 decimal places, not 2.

0.462 = 0.46×0.46 = 0.2116.

When taking the power of a decimal, first count the number of decimal places in the base number, as when multiplying decimals (see Decimal Multiplication. Next, multiply that number by the exponent. This will be the total number of decimal places in the answer. Then, take the power of the base number with the decimal point removed. Finally, insert the decimal point at the correct place, calculated in the second step.


Example 1: 1.54 = ?

1. There is 1 decimal place and the exponent is 4. 1×4 = 4 .
2. 154 = 50, 625 .
3. Insert the decimal point 4 places to the right. 1.54 = 5.0625 .

Example 2: 0.043 = ?

1. There are 2 decimal places and the exponent is 3. 2×3 = 6 .
2. 43 = 64 = 000064 .
3. Insert the decimal point 6 places to the right. 0.043 = 0.000064 .

As we can see, decimals less than 1 with large exponents are generally very small.

Powers of Fractions

The meaning of (3/4)3 is (3/4)×(3/4)×(3/4) , or three-fourths of three-fourths of three-fourths. As shown in the SparkNote on Fractions, when we multiply fractions together, we multiply their numerators together and we multiply their denominators together. To evaluate (3/4)3 = (3/4)×(3/4)×(3/4) , we multiply 3×3×3 , or 33 , to get the numerator and we multiply 4×4×4 , or 43 , to get the denominator. Thus, (3/4)3 = (33)/(43) .

To take the power of a fraction, take the power of the numerator to get the numerator, and take the power of the denominator to get the denominator. To take the power of a mixed number, convert the mixed number into an improper fraction and then proceed as above.


Examples:

I. (5/2)4 = (54)/(24) = 625/16

II. (- 3/4)2 = ((- 3)2)/(42) = 9/16

III. (1/(- 7))3 = (13)/((- 7)3) = 1/(- 343) = - 1/343

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