REAL NUMBERS(LECTURE 9)

Laws of exponents for real numbers 

Good morning boys................

In the previous classes you have already experienced the method of rationalisation ,representation as well as simplification of real numbers.

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1.      take your SET-A Mathematics Register

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6.      Pending Queries from yesterday (if any).......

7.      Please write the learning outcomes as mentioned below

Today's learning outcome is how to apply laws of exponents to simplify real numbers.

Exponents are the repeated multiplication of any number. The number, 1,000,000 can be written as:

1,000,000 = 10 × 10 × 10 × 10 × 10 × 10 = 106, where 6 is the power of 10.

Definition of Powers and Exponents

The exponent is a simple but powerful tool. It tells us how many times a number should be multiplied by itself to get the desired result. Thus any number ‘a’ raised to power ‘n’ can be expressed as:

Here a is any number and n is a natural number.

an is also called the nth power of a.

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

 ‘a’ is multiplied ‘n’ times thereby exponentiation is the shorthand method of repeated multiplication.

1. Product law

According to the product law of exponents when multiplying two numbers that have the same base then we can add the exponents

am × an= a m+n

where a, m and n all are natural numbers. Here the base should be the same in both the quantities. For example,

• 2³ × 24 = 27

2. Quotient Law

According to the quotient law of exponents, we can divide two numbers with the same base by subtracting the exponents. In order to divide two exponents that have the same base, subtract the power in the denominator from the power in the numerator.

am ÷ an = a m-n

where a, m and n all are natural numbers. Here the base should be the same in both the quantities. For example,

• 25 ÷ 23 = 2²
• p6 ÷ p2 = p 6 – 2  = p 4

3. Power Law

According to the power law of exponents if a number raise a power to a power, just multiply the exponents

(am)n = am×n

Here there is one base a  and two powers m and n. For example, ( 53 )2 = 53×2 = 56

Important Points to Remember on Exponent Rules

• 1an = a-n. A non zero base raised to a negative exponent is equal to the reciprocal of the base raised to the positive exponent or 1a−n = a+n
• a0 = 1. This says that anything raised to the zero power is 1. For example, 50= 1, (1000) 0 = 1
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we can extend these laws of exponents to the cases where the base is a positive real number and the exponents are rational numbers

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Question 1

The value of {2 - 3(2 - 3)³}³

Solution 

{2 - 3(2 - 3)³}³ 

= {2 - 3(-1)³}³

= {2 - 3(-1)}³

= {2 - (-3)}³ 

= {2 + 3}³

= {5)³ 

= 5³ 

= 125

Question 2

The value of x - y^x-y when x = 2 and y = -2

Solution 

x = 2, y = -2

x - y = 2 - (-2) = 2 + 2 = 4

Now x - y^x-y = 2 - (-2)⁴ = 2 - 16 = -14

Question 3

The product of the square root of x with the cube root of x

Today you have double period, revise each and every examples from NCERT. Then start your revision .
TODAY'S HOME WORK 

That's all for today....

Revise the chapter well and be ready for a small TEST ON MONDAY.

Good morning and thank you boys.