Exponents and Powers

Exponents and powers are methods for representing extremely large or extremely small numbers in a simplified manner. For example, if we need to express 3 x 3 x 3 x 3 in a straightforward manner, we can write it as 3⁴, where 4 is the exponent and 3 is the base. The entire expression 3⁴ is said to represent power. The laws of exponents can also be learned here.

Power is a mathematical phrase that describes the multiplication of the same number or factor over and over again. The number of times the base is multiplied by itself determines the exponent’s value

What do you mean by Exponent?

The exponent represents the number of times a number is multiplied by itself.. When the number 8 is multiplied by itself n times, the result is:

8 x 8 x 8 x 8 x …..n times = 8n

In the above equation, 8n is written as 8 multiplied by n. As a result, exponents are frequently referred to as power or indices.

Examples:

• 2 x 2 x 2 x 2 = 24

• 5 x 5 x 5 = 53

• 10 x 10 x 10 x 10 x 10 x 10 = 106

Exponents in a General Form

The exponent is a straightforward but effective tool. It tells us how many times to multiply a number by itself to reach the desired result. As a result, every integer ‘a’ raised to the power ‘n’ can be written as:

Powers And Exponents

A can be any number, and n can be any natural number.

The nth power of a is an 

The base is ‘a,’ and the exponent, index, or power is ‘n.’

Exponentiation is the shorthand way of repeated multiplication in which ‘a’ is multiplied ‘n’ times.

Laws of Exponents

The powers of exponents are used to demonstrate the laws of exponents.

• Bases – multiplying like numbers, adding exponents while keeping the base constant. (Law of Multiplication)

• Bases – increase it to a higher power, and multiply the exponents while keeping the base the same.

• Divide the like ones using the formula ‘Numerator Exponent – Denominator Exponent’ while keeping the base constant. (Law of Division)

Let ‘A’ be any positive or negative number or integer, and ‘m’ and ‘n’ be positive integers representing the power to the bases, then;

Law of Multiplication

The product of two exponents with the same base and distinct powers equals the base raised to the total of the two powers or integers, according to the multiplication law of exponents.

Am  +A n =Am+n

Law of Division

When two exponents with the same base but different powers are split, the base is increased to the difference between the two powers.

Am ÷ An  = Am / An  = Am-n

Law of Negative Exponents

If a base has a negative power, the reciprocal with a positive power or integer to the base will be produced.

A -m  = 1/Am

Exponents’ Rules

The laws follow the exponentiation rules. Let’s take a look at them and provide some context.

If ‘a’ and ‘b’ are integers and ‘m’ and ‘n’ are power values, then the rules for exponents and powers are as follows:

i) A0 = 1

If the power of any number is zero, the ensuing output is unity or one, according to this rule.

50 = 1 is an example.

ii) (Am)n = A(mn)

‘A’ to the power of ’m’ to the power of ‘n’ equals ‘a’ to the power of the product of ’m’ and ‘n’.

(52 )3= 52×3  is an example

iii) Am × Bm =(AB)m

The product of ‘a’ raised to the power of ‘m’ and ‘b’ raised to the power of ‘m’ is equal to the product of ‘a’ and ‘b’ raised to the power of ‘m’ as a whole.

5² 6² =(5 x 6)² is an example.

iv) Am/Bm = (A/B)m

The division of ‘a’ by ‘b’ whole raised to the power ‘m’ is the same as the division of ‘a’ by ‘b’ whole raised to the power ‘m’.

Example: 5²/6² = (5/6)²

Exponents with Negative Values

   A negative exponent indicates how many times the base’s reciprocal must be multiplied. For instance, if a-n is supplied, it can be expanded to 1/an. It indicates we have to multiply 1/an’ times the reciprocal of a. When writing fractions using exponents, negative exponents are utilized.

Fractions and Exponents

  When the exponent of a number is a fraction, it is referred to as a fractional exponent. Fractional exponents include square roots, cube roots, and the nth root. The square root of the base is a number having a power of 1/2. Similarly, the cube root of the base is an integer with a power of 1/3.

Exponents of a Decimal

A decimal exponent is one in which the exponent of a number is expressed in decimal form. Because evaluating the correct answer of any decimal exponent is a little challenging, we use an approximation in such circumstances. To solve decimal exponents, convert the decimal to fraction form first. 4 1.5 for example, can be expressed as 4 3/2 which can be simplified even further to achieve the final result of 8.

Exponentiation in Scientific Notation

The traditional form of writing is very big or very small numbers in scientific notation. In this, decimal and powers of ten are used to write numbers. When a number from 0 to 9 is multiplied by a power of ten, it is considered to be written in scientific notation. The power of 10 is a positive exponent when a number is greater than one; when a number is less than one, the power of ten is negative. Let’s look at how to write numbers with exponents in scientific notation:

• Step 1: After the first digit of the number from the left, add a decimal point. We don’t need to put decimal if a number has only one digit, omitting zeros.

• Step 2: Multiply that amount by a power of ten so that the power equals the number of times the decimal point is shifted.

Tricks and Tips:

If the exponent of a fraction is negative, the reciprocal of the fraction is used to make the exponent positive. (a/b)-m = 1/(b/a)m,