Pythagorean Triplet
A Pythagorean triplet consists of three positive integers a, b, and c, such that \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\].
Pythagorean theorem states that, in any right triangle, the sum of squares of base and height is equal to the square of its hypotenuse. Pythagorean triplets describe a relation among three sides of a right triangle. For every natural number n > 1, we have the Pythagorean triplet is given by \[(2n,{{n}^{2}}-1,{{n}^{2}}+1)\]
Let n = 3, then the corresponding Pythagorean triplet is obtained as
\[\text{2n}=\text{2}\times \text{3}=\text{6}\]
\[{{\text{n}}^{2}}-1={{\text{3}}^{2}}-1=8\]
\[{{\text{n}}^{2}}+1={{\text{3}}^{2}}+1=10\]
Hence 6, 8, 10 are Pythagorean triplets.
Square of Negative Numbers
Square of a negative number is always positive. Some example of square of negative numbers are given below:
\[{{(-a)}^{2}}=-a\times -a={{a}^{2}}\]
Numbers between Square Numbers
In general there are 2n non-perfect square numbers between the squares of any two numbers n and n + 1.
For example between 5 and 6 the number of numbers is \[{{6}^{2}}-{{5}^{2}}=36-25=11\]
Thus there are 10 non square numbers which is one less than the difference of the square of two numbers which is equal to 2n i.e. \[\text{2}\times \text{5}=\text{1}0\]. 
Properties of Square Number
A Perfect Square Number can only end with Digits 0, 1, 4, 5, 6 and 9,
Methods to Find Square of a Number
and represents digits before 25. For example the square of 65 can be calculated by 
Thus square of 65 is equal to 4225.
. For example the square of 70 is 4900, where n = 49 & m = 7.
Example: Calculate the square of 57. AA = 25 + 7 = 32 and BB = 49, it means 
.
. It follows that square roots of even square numbers are even and square roots of odd square numbers are odd. 
Square of a Number
The square of a number is the product of the number with itself. Thus for a given number 'a', \[{{a}^{2}}=a\times a\].
Thus to find the square of a number, we have to multiply the number with itself.
Perfect Square of a Number
A natural number is said to be its perfect square, if it can be written as the square of the factors of natural number. Some of the numbers which are perfect squares are:
Squares From \[{{1}^{2}}\] TO \[{{9}^{2}}\]
1 Squared = \[{{1}^{2}}\] = \[\text{1}\times \text{1}\] = 1
2 Squared = \[{{2}^{2}}\] = \[2\times 2\] = 4
3 Squared = \[{{3}^{2}}\] = \[3\times 3\] = 9
4 Squared = \[{{4}^{2}}\] = \[4\times 4\] = 16
5 Squared = \[{{5}^{2}}\] = \[5\times 5\] = 25
6 Squared = \[{{6}^{2}}\] = \[6\times 6\] = 36
7 Squared = \[{{7}^{2}}\] = \[7\times 7\] = 49
8 Squared = \[{{8}^{2}}\] = \[8\times 8\] = 64
9 Squared = \[{{9}^{2}}\] = \[9\times 9\] = 81
The perfect squares are the squares of the whole numbers.
| Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | more...
 Laws of Exponent
There are various laws of exponents. They are laws of addition, laws of multiplication and laws of division.
(i) \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
(ii) \[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\]
(iii) \[{{a}^{m}}\times {{b}^{m}}={{(a\times b)}^{m}}\]
(iv) \[{{\left[ {{\left( \frac{a}{b} \right)}^{n}} \right]}^{m}}={{\left( \frac{a}{b} \right)}^{mn}}\]
(v) \[{{\left( \frac{a}{b} \right)}^{-n}}={{\left( \frac{b}{a} \right)}^{n}}\]
(vi) \[{{\left( \frac{a}{b} \right)}^{0}}=1\]
(vii) \[{{(ab)}^{n}}={{a}^{n}}{{b}^{n}}\]
 Important Points to keep in Mind
(i) \[{{2}^{3}}{{2}^{5}}={{2}^{3+5}}={{2}^{8}}\]
(ii) \[{{w}^{2}}{{w}^{3}}={{w}^{5}}\]
(iii) \[x{{y}^{2}}{{x}^{3}}{{y}^{3}}{{x}^{4}}{{y}^{4}}={{x}^{8}}{{y}^{9}}\]
While working with exponents there are certain rules that we need to remember.
\[{{\text{4}}^{\text{2}}}\times {{\text{4}}^{\text{5}}}=4\text{7}\]
It means:\[\text{4}\times \text{4}\times \text{4}\times \text{4}\times \text{4}\times \text{4}\times \text{4}\] or \[\text{4}\text{.4}\text{.4}\text{.4}\text{.4}\text{.4}\text{.4}\]
Add the exponent, if base are same.
Uses of Exponents
The exponents can be used for various purposes such as comparing large and small numbers, expressing large and small numbers in the standard forms. It is used to express the distance between any two celestial bodies which cannot be expressed in the form of normal denotion. It is also useful in writing the numbers in scientific notation. The size of the microorganisms is very-very small and it cannot be written in normal denotion and can easily be expressed in exponential form.
Radicals Expressed with Exponents
Radicals are the fractional exponents of any number. Index of the radical becomes the denominator of the fractional power.
\[\sqrt[n]{a}=\frac{1}{{{a}^{n}}}\]
i.e. \[\sqrt{9}=\sqrt[2]{9}={{9}^{\frac{1}{2}}}=3\]
Express \[\sqrt[\mathbf{3}]{\mathbf{2}}\,\sqrt[\mathbf{4}]{\mathbf{2}}\] as a Single Radical Term
Let us convert the radicals to exponential expressions, and then apply laws of exponent to combine the factors:
\[\sqrt[3]{2}\,\,\sqrt[4]{2}={{2}^{\frac{1}{3}}}\,\,{{2}^{\frac{1}{4}}}={{2}^{\frac{1}{3}+\frac{1}{4}}}={{2}^{\frac{7}{12}}}=\sqrt[12]{{{2}^{7}}}\]
Simplify \[\frac{\sqrt{5}}{\sqrt[3]{5}}\]
Solution:
\[\frac{{{5}^{\frac{1}{2}}}}{{{5}^{\frac{1}{3}}}}={{5}^{\frac{1}{2}.\frac{1}{3}}}={{5}^{\frac{1}{6}}}\]
\[{{\left( \frac{2}{3} \right)}^{4}}=\left( \frac{2}{3} \right)\times \left( \frac{2}{3} \right)\times \left( \frac{2}{3} \right)\times \left( \frac{2}{3} \right)\]
Solution:
\[=\frac{2\times 2\times 2\times 2}{3\times 3\times 3\times 3}=\frac{{{2}^{4}}}{{{3}^{4}}}=\frac{16}{81}\]
Expand: \[{{\left( \frac{x}{10} \right)}^{5}}\]
Solution: Raising the top and bottom numbers to the power of 5 gives:
\[{{\left( \frac{x}{10} \right)}^{5}}=\frac{{{x}^{5}}}{{{10}^{5}}}=\frac{{{x}^{5}}}{100000}\]
\[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
\[{{({{a}^{m}})}^{n}}={{a}^{mn}}\]
\[{{(ab)}^{n}}={{a}^{n}}{{b}^{n}}\]
\[{{\left( \frac{a}{b} \right)}^{n}}=\frac{{{a}^{n}}}{{{b}^{n}}}\]
\[{{a}^{0}}=1\]
\[{{a}^{-n}}=\frac{1}{{{a}^{n}}}\]
more...  Exponents
Any number of the form \[{{a}^{n}}\], where n is a natural number and "a" is a real number is called the exponents. Here n is called the power of the number a. Power may be positive or negative.
For any rational number \[{{\left( \frac{a}{b} \right)}^{n}}\], n is the power of the rational number.
\[{{\left( \frac{a}{b} \right)}^{n}}={{\left( \frac{a}{b} \right)}^{n}}=\frac{a}{b}\times \frac{a}{b}\times \frac{a}{b}\times \frac{a}{b}\times \frac{a}{b}\times ----\times \frac{a}{b}\] (n-times)\[=\frac{{{a}^{n}}}{{{b}^{n}}}\]
\[{{x}^{n}}=x\times x\times x\times x\times ---\times x\](n-times) and
\[{{x}^{-n}}=\frac{1}{x\times x\times x\times x\times x\times ---\times x}\]
Also, \[{{x}^{0}}=1;\] \[{{x}^{1}}=x;\] \[{{x}^{-1}}=\frac{1}{x}\] and \[{{x}^{-n}}=\frac{1}{{{x}^{n}}}\]
\[{{3}^{0}}=1\]
\[{{3}^{1}}=3\]
\[{{3}^{-1}}=\frac{1}{3}\]  Variations
If the two quantities are related with each other than change in one quantity will produce the corresponding change in the other quantity. The variation may be that if we increase or decrease the one quantity then other quantity may also increase or decrease and vice-versa. If increase in one quantity results in increase in other quantity then it is called as direct variation and if reverse happens then it is called as indirect variation. For example increase in the cost with the increase in quantity is a direct variation whereas decrease in the time taken for a work if we increase the number of worker then it is a inverse variation.  Direct Variation
Two quantities are said to varies directly if increase in one quantity results the increase in other.
(i) The cost of articles varies directly as the number of articles increases.
(ii) The distance covered by a moving object varies directly as its speed increases or decreases. (It means if speed increases then the more distance covered in the same time).
(iii) The work done varies directly as the number of men increases.
(iv) The work done varies directly as the working time increases.  Inverse Variation
Two quantities are said to be vary inversely if increase in one quantity results in decrease in the other quantity and vice versa.
(i) The time taken to finish a piece of work varies inversely as the number of men at work varies, (more men take less time to finish the job)
(ii) The speed varies inversely as the more time taken to cover a distance (more is the speed less is the time taken to cover a distance.
 If 4 women or 3 men earn Rs. 960 in a day, then the earning of 11 women and 7 men in a day will be:
(a) Rs. 4880
(b) Rs. 2200
(c) Rs. 1860
(d) Rs. 1480
(e) None of these
Answer: (a)
Explanation:
One day earning of 4 women or 3 men = Rs. 480
Therefore, one day earning of 1 women or 1 men \[=Rs.\frac{960}{4}or\,Rs.\frac{960}{3}\]
One day earning of 11 women or 7 men \[=Rs.\frac{960}{4}\times 11\,or\,Rs.\frac{960}{3}\times 7\]
= Rs. 2640 or Rs. 2140
Therefore, total earning of 11 women and 7 men for one day is Rs. 2440.
A transport company transports goods from one place to another and charges certain cost for it. A merchant asked to transport 2 quintals of rice from one city to another at a distance of 125 km. The transportation cost of 160 kg of rice from one city to another is Rs. 60. Find the amount the merchant has to pay for the transportation of the entire 2 quintals of rice.
(a) Rs. 75
(b) Rs. 70
(c) Rs. 100
(d) Rs. 125
(e) None of these
Answer: (a)
Harry wants to mix the flour of two different rates so that he can sell at the rate he wants. In what proportion he must mixes the flour at Rs. 16.6 per kg with a flour at Rs. 16.45 per kg so that the mixture can be sold at the rate of Rs. 16.54 per kg.
(a) 1:3 more...  Introduction
An algebraic expression is an expression in one or more variables having many terms. Depending on the number of terms it may be monomials, binomials, trinomials or polynomials. Like in the case of real numbers we can also operate the algebraic expression. Previously we have learnt to add and subtract the algebraic expression. In this chapter we will learn, how to multiply or divide the algebraic expression. We will also learn how to find the linear factors of the algebraic expression as in the case of real numbers. The constants multiplied with the variables in the algebraic expression are called the coefficient of the terms. The coefficient may be positive or negative.
Concept of Monomial, Binomial and Trinomial
 Monomials
The polynomial having one term is called monomial.
\[{{x}^{5}},7x,9xy\] are monomials as they contain only one term.
Binomial
The polynomial which contains two terms is called binomials.
\[\text{4a}+\text{3b},\text{ 2y}+\text{3y}\] etc. are binomials because they contain two terms.
Trinomial
A trinomial is a polynomial containing three terms.
\[3x+\text{5y}+\text{7z},\text{ 2a}+\text{6b}+\text{7c}\] are the polynomials containing three terms.  Addition and Subtraction of Algebraic Expressions
While adding or subtracting the algebraic expressions we add or subtract the like terms of the expression. While adding or subtracting the like terms of the algebraic expression we add or subtract the coefficients of the algebraic expression. But in case of multiplication or division we normally multiply or divide each term of one expression with the each term of the other expression.
Add: \[3x+\text{5y}+\text{8z}\] and \[8x+100\text{y-18z}\]
Solution:
\[=3x+5y+8z+8x+100y-18z\]
\[=(3+8)x+(5+100)y+(8-18)z\]
\[=11x+105y-10z\]
 Like Terms and Unlike Terms
The terms having same order of variables are called like terms and the terms which a do not have same order of variables are called unlike terms.
\[7x,-14x,25x\] are like terms while \[8x,9xy,78zx\] are unlike terms. Current Affairs CategoriesArchive
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