**Exponents and Scientific Notation :**

Here we are going to see the basic concept of exponents and scientific notation.

The exponent of a number says, how many times the number has to be multiplied by itself.

For example,

5^{3} = 5 x 5 x 5

In words, 5^{3} could be called as 5 to the power 3 or 5 cube.

10 | |

10 | |

If we have 1/2 in the power, we can simply write the base inside the radical or square root. If we have 1/3 in the power,we can simply write the base inside the cube root |

If the power goes from one side of equal sign to the other side, it will become its reciprocal. That is x = 4 |

The other names of exponent are index and power.

**Law 1 : **

x^{m} ⋅ x^{n} = x^{m+n}

**Example :**

3^{4} ⋅ 3^{5} = 3^{4+5}

3^{4} ⋅ 3^{5} = 3^{9}

**Law 2 : **

x^{m} ÷ x^{n} = x^{m-n}

**Example :**

3^{7} ÷ 3^{5} = 3^{7-5}

3^{7} ÷ 3^{5} = 3^{2}

**Law 3 : **

(x^{m})^{n} = x^{mn}

**Example :**

(3^{2})^{4} = 3^{(2)(4)}

(3^{2})^{4} = 3^{8}

**Law 4 : **

(xy)^{m} = x^{m} ⋅ y^{m}

**Example :**

(3 ⋅ 5)^{2} = 3^{2} ⋅ 5^{2}

(3 ⋅ 5)^{2} = 9 ⋅ 25

(3 ⋅ 5)^{2} = 225.

**Law 5 : **

(x / y)^{m} = x^{m} / y^{m}

**Example :**

(3 / 5)^{2} = 3^{2} / 5^{2}

(3 / 5)^{2} = 9 / 25

**Law 6 : **

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

**Example :**

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

3^{-2} = 1 / 9

**Law 7 : **

x^{0} = 1

**Example :**

3^{0} = 1

**Law 8 : **

x^{1} = x

**Example :**

3^{1} = 3

**Law 9 : **

x^{m/n} = y -----> x = y^{n/m}

**Example :**

x^{1/2} = 3

x = 3^{2/1}

x = 3^{2}

x = 9

**Law 10 : **

(x / y)^{-m} = (y / x)^{m}

**Example :**

(5 / 3)^{-2} = (3 / 5)^{2}

(5 / 3)^{-2} = 3^{2} / 5^{2}

(5 / 3)^{-2} = 9 / 25

**Law 11 : **

a^{x} = a^{y} -----> x = y

**Example :**

3^{m} = 3^{5} -----> m = 3

**Law 12 : **

x^{a} = y^{a} -----> x = y

**Example :**

k^{3} = 5^{3} -----> k = 5

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.

To convert a number into scientific notation, first we have to identify where the decimal point and non zero digit come.

There are two cases in it.

**Case 1 : **

To move the decimal point to the left, we have to count number of digits as explained in the example given below.

According to the example given above, we have to move the decimal point 3 digits to the left and exponent of 10 should be 3 (positive integer)

When we do so, we get the scientific notation of the given number.

**Hence, 2301.8 = 2.3018 x 10³**

**Case 2 : **

To move the decimal point to the right, we have to count number of digits as explained in the example given below.

According to the example given above, we have to move the decimal point 5 digits to the right and exponent of 10 should be -5 (negative integer)

When we do so, we get the scientific notation of the given number.

**Hence, 0.000023 = 2.3 x 10****⁻****⁵**

**Important Note: **

If we don't find decimal point at anywhere of the given number, we have to assume that there is decimal point at the end of the number.

**For example, **

**2300000 -------------> 2300000.**

Here, the non zero digit comes first and decimal point comes next. So we have to apply case 1 to convert this number into scientific notation.

If you would like to have practice problems on exponents,

If you would like to have practice problems on scientific notation,

After having gone through the stuff above, we hope that the students would have understood the basic stuff about exponents and scientific notation.

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