**Zero and Negative Exponents Properties and Scientific Notation :**

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

**Zero Exponent Rule :**

**Anything to the power zero is 1.**

**Examples :**

a^{0} = 1

5^{0} = 1

(-7)^{0} = 1

**Negative Exponents Properties :**

In a number, if the exponent is negative, it can be made it as positive by taking the reciprocal of the base.

Examples :

2^{-3} = (1/2)^{3}

(1/7)^{-5} = 7^{5}

(2/7)^{-8} = (7/2)^{8}

Scientific notation is a standard way of writing very large and very small numbers so that they’re easier to both compare and use in computations.

Every number in the scientific notation must be in the form of

**a x 10 ^{n}**

**where ****1 ****≤ a < 10** and **n** must be a positive or negative integer.

**Step 1 :**

Adjust the decimal point such that there is only one non zero digit on the left side of the decimal point.

**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 point is shifted to the left, the exponent n will be positive. If the decimal point is shifted to the right, the exponent n will be negative.

To convert a number from scientific notation to standard notation, first we have to notice the exponent of 10 in scientific notation.

If the exponent of 10 is positive, we have to move the decimal point to the right.

For example, if you have 10^{3}, you have to move the decimal point 3 digits to the right.

If the exponent of 10 is negative, we have to move the decimal point to the right.

For example, if you have 10^{-5}, you have to move the decimal point 5 digits to the left.

**Example 1 :**

Simplify :

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

**Solution :**

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

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

^{ }(2x^{2})^{-4 }= 1 / (16x^{8})

**Example 2 :**

Simplify :

(x^{3})^{0}

**Solution :**

(x^{3})^{0 }= 1

**Example 3 :**

Simplify :

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

**Solution :**

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

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

(r^{4}/5)^{-2 }= 25 / r^{8}

**Example 4 :**

Simplify :

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

**Solution :**

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

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

x^{-1 }/ 4x^{4 }= 1 / (4x^{5})

**Example 5 :**

Express 9781 in scientific notation.

**Solution :**

Here we don't find decimal point in 9781. So we have to assume that there is decimal point at the end .

Then,

9781 ------> 9781.

Here non zero digit comes first and decimal point comes next.

We have to move the decimal point to the left.

No. of digits between the 1st non zero digit and the decimal point is 3.

So, the decimal point has to be moved three digits to the left and exponent of 10 should be 3 (positive integer)

9781 = 9.781 x 10^{3}

Hence, the scientific notation of 9781 is

5.4 x 10^{6}

**Example 6 :**

Express 0.000432078 in scientific notation.

**Solution :**

Here decimal point comes first at non zero digit comes next.

We have to move the decimal point to the right.

No. of digits from the decimal point up to the first non zero digit is 4.

So, the decimal point has to be moved 4 digits to the right and exponent of 10 should be -4 (negative integer)

Hence, the scientific notation of 0.000432078 is

4.32078 x 10^{-4}

**Example 7 :**

Write the following numbers in decimal form.

5.236 x 10^{5}

**Solution :**

Here, the exponent of 10 is positive 5. So we have to move the decimal point five digits to the right.

In 5.236, we have only three digits after the decimal point.

So, we have to add two zeros to move the decimal point five digits to the right.

Therefore, the standard form of 5.236 x 10^{5} is

523600

**Example 8 :**

Write the following numbers in decimal form.

9.36 x 10^{-9}

**Solution :**

Here, the exponent of 10 is negative 9. So we have to move the decimal point nine digits to the left.

In 9.36, we have only one digit before the decimal point.

So, we have to add eight zeros to move the decimal point nine digits to the left.

Therefore, the decimal form of 9.36 x 10^{-9} is

0.00000000936

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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