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**Exponents**

We can write large numbers in a short form using exponents.

For example: 10,000 = 10 × 10 × 10 × 10 = 10^{4}

Here, ‘10’ is called the base and ‘4’ the exponent. The number 10^{4} is read as 10 raised to the power of 4 or simply as the fourth power of 10.

10^{4} is called the exponential form of 10,000.

(1)^{any natural number} = 1

(-1)^{an odd natural number} = -1

(-1)^{an even natural number} = +1

a^{m} × a^{n} = a^{m+n}, where m and n are whole numbers and a (≠ 0) is an integer.

This formula can be used to write answers to above questions.

For any non-zero integer a,

a^{m} ÷ a^{n} = a^{m-n} where m and n are whole numbers and m > n.

For any non-zero integer a,

(a^{m})^{n} = a^{mn} (where m and n are whole numbers)

For any non-zero integer a

a^{m} × b^{m} = (ab)^{m} (where m is any whole number)

(where m is a whole number; a and b are any non-zero integers)

a^{0} = 1 (for any non-zero integer a)

Any number (except 0) raised to the power (or exponent) 0 is 1.

**Decimal Number System**

10,000 = 10^{4}

1000 = 10^{3}

100 = 10^{2}

10 = 10^{1}

1 = 10^{0}

We can write the expansion of a number using powers of 10 in the exponent form.

**Expressing Large Numbers in the Standard Form**

Large numbers can be expressed conveniently using exponents. Such a number is said to be in standard form if it can be expressed as k × 10^{m}, where 1 ≤, k < 10 and m is a natural number.

Note that, one less than the digit count (number of digits) to the left of the decimal point in a given number, is the exponent of 10 in the standard form.

For any rational number a and positive integer n, we define a^{n} as a × a × a × …… × a (n times). a^{n} is known as the nth power of a and is read as ‘a raised to the power n’. The rational a is called the base and n is called the exponent or power.

e.g. 10,000 = 10 × 10 × 10 × 10 = 10^{4}.

10 is the base and 4 is the exponent.

Multiplying Powers with the Same Base: If a is any non-zero integer and whole numbers are m and n, then a^{m} × a^{n} = a^{m+n}

e.g. 2^{4} × 2^{2}

a = 2, m = 4, n = 2

2^{4} × 2^{2} = 2^{4+2} = 2^{6}

Dividing Powers with the Same Base: If a is any non-zero integer and m, n are the whole number, then a^{m} ÷ a^{n} = a^{m-n}

e.g. 2^{4} ÷ 2^{2}

a = 2, m = 4, n = 2

2^{4} ÷ 2^{2} = 2^{4-2} = 2^{2}

Taking Power of a Power: If a is any non-zero integer and m, n are whole numbers, (a^{m})^{n} = a^{mn}

e.g. (6^{2})^{4}

a = 6, m = 2, n = 4

(6^{2})^{4} = (6)^{2×4} = 6^{8}.

Multiplying Powers with the Same Exponents: If a, b are two non-zero integers and m is any whole number, then

a^{m} × b^{n} = (a × b)^{m}

e.g. 2^{3} × 3^{3}

a = 2, b = 3, m = 3

2^{3} × 3^{3} = (2 × 3)^{3} = 6^{3}.

Dividing Powers with the Same Exponents: If a, b are two non-zero integers and m is a whole number, then

Numbers with Exponent Zero: If a be any non-zero integer, then, a^{0} = 1

Numbers with Negative Exponent: If a is any non-zero integer, then a^{-1} = \(\frac { 1 }{ a }\)

e.g. 2^{-5} = \(\frac { 1 }{ { 2 }^{ 5 } }\)

In decimal number system, the exponents of 10 start from a maximum value and go on decreasing from the left to right upto 0.

e.g. 45672 = 4 × 10000 + 5 × 1000 + 6 × 100 + 7 × 10 + 2 × 1

= 4 × 10^{4} + 5 × 10^{3} + 6 × 10^{2} + 7 × 10^{1} + 2 × 10^{0}

It is called expanded form of a number.

Any number can be expressed as a decimal number between 1.0 and 10.0 including 1.0 multiplied by a power of 10. Such a form of a number is called its standard form.

e.g. 56782 = 5.6782 × 10000 = 5.6782 × 10^{4}.

It is the standard form of 56782.