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Exponents

In mathematics, an exponent is a notation used to indicate how many times a number is multiplied by itself. The exponent is usually written as a superscript number to the right of the base number and indicates how many times the base is multiplied by itself.

For example, 54 (read as "5 raised to the power of 4") means 5 multiplied by itself 4 times, which is equal to 625. The number 5 is called the “base” and 4 is called the “exponent” or “power”.

Standard Form of Exponents

The general form of x^{n}, where x is any real number and n is a positive integer, is the product of x multiplied by itself n times.

-> x is the "base". -> n is the "exponent". -> x^{n} is pronounced as "x to the power of n" (or) "x raised to n". Generally, x^{n} represents that x has been multiplied by itself n times.

Examples:

1. 2^{3} = 2 x 2 x 2 = 8

Solution: 2^{3} read as “Two raised to the power of three”.

2. 5^{2} = 5 x 5 = 25

Solution: 5^{2} read as “five raised to the power of two”.

3. 3^{4} = 3 x 3 x 3 x 3 = 81

Solution: 3^{4} read as “three raised to the power of four”.

4. 6^{0} = 1

Solution: 6^{0} read as “six raised to the power of zero”.

5. (-2)^{3} = -2 x -2 x -2 = -8

Solution: (-2)^{3} read as “minus two raised to the power of three”.

6. (2/3)^{-2} = 3/2 x 3/2 = 9/4

Solution: (2/3)^{-2} read as “two-thirds raised to the power of negative two”.

7. (42)^{3} = 4 x 4 x 4 x 4 x 4 x 4 = 4096.

Solution: (42)^{3} read as “four raised to the power of two, then raised to the power of three”.

Exponents as a Scientific Notation

Exponents can be expressed in scientific notation since they are often used to represent very large or very small numbers.

In scientific notation, an exponent is written as the product of a base and a power of 10. A base is 10 and the power represents the number of times the base must be multiplied by itself.

For example, 10^{-3} means 1/1000 or 0.001 10^{6} means 1,000,000 or 1 million.

Exponents Form

Positive Exponents

A positive exponent indicates that the base is being multiplied by itself a certain number of times.

For example, 2^{3} means 2 multiplied by itself 3 times (2 x 2 x 2 = 8).

Negative Exponents

A negative exponent indicates that the base is being divided by itself a certain number of times.

For example, 2^{-3} means 1 divided by 2 multiplied by itself 3 times (1/2 x 1/2 x 1/2 = 1/8).

Zero Exponents

A zero exponent indicates that the base is being raised to the power of 0, which equals 1.

For example, 2^{0} = 1.

Reciprocal Exponents

A reciprocal exponent is the reciprocal (inverse) of another exponent.

For example, 2^{-1} is the reciprocal of 2^{-1}, which is 1/2.

Mixed Exponents

A mixed exponent is a combination of different types of exponents.

For example, (2^{3})^{2} means 2 multiplied by itself 3 times, then multiplied by itself 2 more times (8 x 8 = 64).

Laws of Exponents

Several laws or rules of exponents are used to simplify expressions involving powers or exponents. These rules are important in algebra and other branches of mathematics and are useful for performing calculations involving numbers raised to powers.

1. Zero Exponent: When the power of the base is zero. a^{0} = 1 Where a is the base and 0 is the power of an exponent

Example: 2^{0} = 1

2. Product of Powers: When multiplying two powers with the same base, add the exponents.

where m and n are real numbers.

Example: What is the simplification of 6^{5} × 6^{1}?

Solution: 6^{5} × 6^{1} = 6 ^{5+1} = 6^{6}

3. Quotient of Powers: When dividing two powers with the same base, subtract the exponents.

where “a” is a non-zero term and “m” and “n” are integers.

Example: Determine the value when 10^{5}/10^{3}.

Solution: 10^{5}/10^{3} = 10^{5-3} = 10^{2} = 100

So, the value when 10^{5} is divided by 10^{3} is 10^{2}.

4. Power of a Power: When raising a power to another power, multiply the exponents.

Example: Find (2^{3})^{3}.

Solution: (2^{3})^{3} = 2^{9} = 512

5. Power of a Product:

Example: Determine and write the exponential form of 2^{-3} x 5^{-3}.

Example: Simplify the expression and find the value:(7/5)^{3}.

Solution: We can write the given expression as 7^{3}/5^{3} = 343/125

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