What Are Exponents In Number System Class 9?

In the Class 9 Number System, exponents are a fundamental concept used to express repeated multiplication in a concise form.

An expression like ana to the n-th power

𝑎𝑛

consists of two main parts:

• Base (a): The number being multiplied by itself.
• Exponent (n): A smaller number written as a superscript that tells you how many times to multiply the base by itself.

For example, in the expression 535 cubed

53

:

• The base is 5.

• The exponent is 3.

• This means 5×5×5=1255 cross 5 cross 5 equals 125

  5×5×5=125

  .

Exponents are also referred to as "powers" or "indices" and are essential for working with very large or very small numbers, such as in scientific notation.

Laws of exponents for real numbers

The laws of exponents provide rules for simplifying and solving expressions involving powers. In Class 9, these laws are extended to include rational exponents, where the power is a fraction.

Let aa

𝑎

and bb

𝑏

be any real numbers, and mm

𝑚

and nn

𝑛

be rational numbers.

1. Product law

To multiply two exponential expressions with the same base, you add the exponents.

• **Formula:**am×an=am+na to the m-th power cross a to the n-th power equals a raised to the m plus n power

  𝑎𝑚×𝑎𝑛=𝑎𝑚+𝑛

• **Example:**23×24=23+4=272 cubed cross 2 to the fourth power equals 2 raised to the 3 plus 4 power equals 2 to the seventh power

  23×24=23+4=27

2. Quotient law

To divide two exponential expressions with the same base, you subtract the exponent of the denominator from the exponent of the numerator.

• **Formula:**am÷an=am−na to the m-th power divided by a to the n-th power equals a raised to the m minus n power

  𝑎𝑚÷𝑎𝑛=𝑎𝑚−𝑛

• **Example:**56÷52=56−2=545 to the sixth power divided by 5 squared equals 5 raised to the 6 minus 2 power equals 5 to the fourth power

  56÷52=56−2=54

3. Power of a power law

When an exponential expression is raised to another power, you multiply the exponents.

• Formula:(am)n=amnopen paren a to the m-th power close paren to the n-th power equals a raised to the m n power

  (𝑎𝑚)𝑛=𝑎𝑚𝑛

• Example:(32)4=32×4=38open paren 3 squared close paren to the fourth power equals 3 raised to the 2 cross 4 power equals 3 to the eighth power

  (32)4=32×4=38

4. Power of a product law

When a product of bases is raised to an exponent, you distribute the exponent to each base.

• Formula:(ab)m=ambmopen paren a b close paren to the m-th power equals a to the m-th power b to the m-th power

  (𝑎𝑏)𝑚=𝑎𝑚𝑏𝑚

• Example:(2×3)4=24×34open paren 2 cross 3 close paren to the fourth power equals 2 to the fourth power cross 3 to the fourth power

  (2×3)4=24×34

5. Power of a quotient law

When a quotient of bases is raised to an exponent, you distribute the exponent to both the numerator and the denominator.

• Formula:(a/b)m=am/bmopen paren a / b close paren to the m-th power equals a to the m-th power / b to the m-th power

  (𝑎/𝑏)𝑚=𝑎𝑚/𝑏𝑚

• Example:(4/5)3=43/53open paren 4 / 5 close paren cubed equals 4 cubed / 5 cubed

  (4/5)3=43/53

6. Negative exponent law

A negative exponent signifies the reciprocal of the base raised to the positive exponent.

• **Formula:**a−m=1/ama raised to the negative m power equals 1 / a to the m-th power

  𝑎−𝑚=1/𝑎𝑚

• **Example:**4-2=1/42=1/164 to the negative 2 power equals 1 / 4 squared equals 1 / 16

  4−2=1/42=1/16

7. Zero exponent law

Any non-zero number raised to the power of zero is equal to 1.

• **Formula:**a0=1a to the 0 power equals 1

  𝑎0=1

  (where a≠0a is not equal to 0

  𝑎≠0

  )

• **Example:**70=17 to the 0 power equals 1

  70=1

8. Rational exponent law

A fractional exponent indicates both a root and a power.

• **Formula:**am/n=amn=(an)ma raised to the m / n power equals the n-th root of a to the m-th power end-root equals open paren the n-th root of a end-root close paren to the m-th power

  𝑎𝑚/𝑛=𝑎𝑚𝑛√=(𝑎𝑛√)𝑚

• **Example:**642/3=6423=(643)2=42=1664 raised to the 2 / 3 power equals the cube root of 64 squared end-root equals open paren the cube root of 64 end-root close paren squared equals 4 squared equals 16

  642/3=6423√=(643√)2=42=16

Application: Scientific notation

Scientific notation is a practical application of exponents for writing very large or very small numbers. A number is written in the form a×10na cross 10 to the n-th power

𝑎×10𝑛

, where:

• **aa

  𝑎** is the coefficient, which is a number greater than or equal to 1 and less than 10 (1≤|a|<101 is less than or equal to the absolute value of a end-absolute-value is less than 10

  1≤|𝑎|<10

  ).

• 10 is the base.

• **nn

  𝑛** is the exponent, a non-zero integer.

Examples:

• To write a large number like 149,600,000 in scientific notation, you move the decimal point to the left until you have a number between 1 and 10. The number of places moved is the positive exponent.149,600,000=1.496×108149 comma 600 comma 000 equals 1.496 cross 10 to the eighth power

  149,600,000=1.496×108

• To write a small number like 0.0002 in scientific notation, you move the decimal point to the right. The number of places moved is the negative exponent.0.0002=2×10-40.0002 equals 2 cross 10 to the negative 4 power

  0.0002=2×10−4