**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.

**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 10^{n}, where 1 ≤ a < 10 and n is an integer.

**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.

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.

**Example 1 :**

Simplify (2x^{2})^{-4}

**Solution :**

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

(2x^{2})^{-4 }= 1/(2x^{2})^{4 }

^{ }= 1/[2^{4}(x^{2})^{4 }]

^{ }= 1/16x^{8}

**Example 2 :**

Simplify (x^{3})^{0}

**Solution :**

= (x^{3})^{0}

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

**Example 3 :**

Simplify (r^{4}/5)^{-2}

**Solution :**

= (r^{4}/5)^{-2}

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

(r^{4}/5)^{-2 }= (5/r^{4})^{2}

= 5^{2}/(r^{4})^{2}

= 25/r^{8}

**Example 4 :**

Simplify x^{-1}/4x^{4}

**Solution :**

= x^{-1}/4x^{4}

= (1/4) x^{ (-1 + 4)}

= (1/4) x^{ 3}

= 1/4 x^{3}

Hence the answer is 1/4 x^{3}

**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 10^{3}

**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 10^{5}

**Solution :**

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

5.236 x 10^{5 }= 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

- Properties of exponents
- Negative exponent rules
- Evaluate integers raised to the rational exponents
- Simplifying expressions involving rational exponents

After having gone through the stuff given above, we hope that the students would have understood, "Zero and negative exponents properties and scientific notation".

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