Exponents rules

Exponents

Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the “equals” sign in (5)(5)(5) = 53. The “exponent”, being 3 in this example, stands for however many times the value is being multiplied. The thing that’s being multiplied, being 5 in this example, is called the “base”.
This process of using exponents is called “raising to a power”, where the exponent is the “power”. The expression “53” is pronounced as “five, raised to the third power” or “five to the third”.

 

Rational Numbers and its Properties

Q. What is Rational Number ?
Ans. A Rational Number is a real number that can be written as a simple fraction (i.e. as a ratio).
A rational number is a number that can be in the form p/q
where p and q are integers and q is not equal to zero.
So, a rational number can be:
p/q

Where q is not zero

Q. What are Positive rational numbers ?

Ans. A rational number is said to be positive if its numerator and denominator are either both positive integers or both negative integers.
In other words, a rational number is positive, if its numerator and denominator are of the same sign.

Q. What are Negative rational numbers ?

Ans. A rational number is said to be negative if its numerator and denominator are of opposite signs such that, one of them is positive integer and another one is a negative integer.
In other words, a rational number is negative, if its numerator and denominator are of the opposite signs.

Q. what is lowest term of a rational number ?

Ans. A rational expression has been reduced to lowest terms if all common factors from the numerator and denominator have been canceled.
Some solved example

1) Write in lowest form : i) 17 / 79 ii) -60 / 72
Solution : i) 17 / 79

As HCF of 17 and 79 is 1 so 17 / 79 is in the lowest form.

ii) -60 / 72

HCF of 60 and 72 is 12.

-60 ÷ 12 = – 5 and 72 ÷ 12 = 6.

∴ -60 / 72 = – 5 / 6.

Q. What is Standard form of a Rational Number ?

Ans. A rational number is in standard or simplest or lowest form when following two conditions are fulfilled:
• Numerator and denominator have only 1 as its highest common factor.
• Denominator is a positive integer.

Lets study some examples on this –

Example : Is rational number 6/7 is in standard form ?
Solution : This proceeds in the following ways:

First find the HCF of denominator and numerator i.e. 6 & 7 and we get:
HCF of 6 & 7 = 1

Since, Numerator and denominator have only 1 as its highest common factor; so as explained above first condition is fulfilled.

Now check second condition ?

Here we can see that denominator is 7, which is a positive integer. So this fulfills second condition also.

Now, since both the conditions are fulfilled, we can say that given rational number 6/7 is in standard form.

Q. What are Equivalent Rational Numbers ?

Ans. The equivalent rational numbers are numbers that have same value but are represented differently.
*The equivalent rational number defines the equivalence of fractions in math and the equivalent ratio for the numbers.

The ratios of the two numbers have equivalent ratio for the ratios of the other two numbers. Here, we are going to see about the equivalent forms of rational numbers for the two numbers.
Two rational numbers are said to be equivalent if we divide each decimal value will be the same

 

Q. COMPARISON OF RATIONAL NUMBERS ?

Ans. Among the positive rational numbers with the same denominator, the number with the greatest numerator is the largest. It is easy to compare the rational numbers with same denominators.
e.g. 2830 > 2630 > 2130 .

A negative rational number is to the left of zero whereas a positive rational number is to the right of zero on a number line. So, a positive rational number is always greater than a negative rational number.

To compare two negative rational numbers with the same denominator, their numerators are compared ignoring the minus sign. The number with the greatest numerator is the smallest.
e.g. – 710 < – 310 ; – 6 7 < – 4 7

To compare rational numbers with different denominators, they are converted into equivalent rational numbers with the same denominator, which is equal to the LCM of their denominators.

There are unlimited number of rational numbers between two rational numbers. To find a rational number between the given rational numbers, they are converted to rational numbers with same denominators

PROPERTIES OF RATIONAL NUMBERS

Rational numbers are terminating or recurring decimal numbers written in the form of fraction p/q In which ‘p’ and ‘q’ are integers and the denominator ‘q’ not equal to zero.

Let a,b,c be three rational numbers and the properties of rational numbers are given below:

Rational numbers are commutative and associative under addition and multiplication.

Commutative law:
a + b = b + a
a x b = b x a

Associative law:
a + (b + c) = (a + b) + c
a x (b x c) = (a x b) x c

Rational numbers holds true for closure law under addition, subtraction and multiplication.
a + b = a rational number
a – b = a rational number
a x b = a rational number
a/b = not a rational number

Rational numbers have an additive identity of 0 and multiplicative identity of 1.
a + 0 = a
a x 1 = a

Rational numbers holds true for distributive property also.
a + (b x c) = (a + b) x (a + c)
(a + b) x c = (a x c) + (b x c)