ZERO AND NEGATIVE EXPONENTS PROPERTIES AND SCIENTIFIC NOTATION

About "Zero and negative exponents properties and scientific notation"

Zero and negative exponents properties and scientific notation :

Here we are going to discuss about properties of zero and negative exponents and scientific notation.

Zero Exponent Rule :

Anything to the power zero is 1.

Negative exponents properties :

Whenever we have a negative number as exponent and we need to make it as positive, we have to flip the base that is write the reciprocal of the base and we can change the negative exponent as positive exponent.

Let us see some examples to understand this concept.

Integer with negative exponent:

Fractions with negative exponents

Scientific notation :

A number N is in scientific notation when it is expressed as the product of a decimal number between 1 and 10 and some integral power of 10.

N = a x 10n, where 1  a < 10 and n is an integer.

Converting a number to scientific notation

Step 1 :

Move the decimal point so that there is only one non - zero digit to its left.

Step 2 :

Count the number of digits between the old and new decimal point. This gives n, the power of 10.

Step 3 : 

If the decimal is shifted to the left, the exponent n is positive. If the decimal is shifted to the right, the exponent n is negative.

Converting Scientific Notation to Decimal Form

Often, numbers in scientific notation need to be written in decimal form. To convert scientific notation to integers we have to follow these steps.

Step 1 :

Write the decimal number.

Step 2 :

Move the decimal point the number of places specified by the power of ten: to the right if positive, to the left if negative. Add zeros if necessary.

Step 3 :

Rewrite the number in decimal form.

Zero and negative exponents properties and scientific notation - Examples

Example 1 :

Simplify (2x2)-4

Solution :

To convert the negative power as positive, we need to take its reciprocal.

(2x2)-4   =  1/(2x2)

  =  1/[24(x2)]

  =  1/16x8

Example 2 :

Simplify (x3)0

Solution :

  =  (x3)0

Since we have power zero, the answer must be 1.

Example 3 :

Simplify (r4/5)-2

Solution :

=  (r4/5)-2

To convert the power as positive, we need to take its reciprocal. 

(r4/5)-2  =  (5/r4)2

=  52/(r4)2

=  25/r8

Example 4 :

Simplify x-1/4x4

Solution :

  =  x-1/4x4

  =   (1/4) x (-1 + 4)

=   (1/4) x 3

=   1/4 x3

Hence the answer is 1/4 x3

Example 5 :

Express 9781 in scientific notation.

Solution :

In integers, the decimal point at the end is usually omitted. 

9 7 8 1 .

The decimal point is to be moved 3 places to the left of its original position. So the power of 10 is 3.

9781  =  9.781 x 103

Example 6 :

Express 0 . 000432078 in scientific notation.

Solution :

0 . 0 0 0 4 3 2 0 7 8

The decimal point is to be moved four places to the right of its original position. So the power of 10 is – 4

0 . 0 0 0 4 3 2 0 7 8  =  4.32078 x 10-4

Example 7 :

Write the following numbers in decimal form.

5.236 x 105

Solution :

Since we have positive power, we need to move the decimal point 5 digits to the right side.

5.236 x 105  =  523600

Example 8 :

Write the following numbers in decimal form.

9.36 x 10-9

Solution :

Since we have negative power, we need to move the decimal point 9 digits to the left side.

9.36 x 10-9  =  0.00000000936

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