Found 225 Articles for Class 8

Given:Given polynomials are $42x^2yz$ and $63x^3y^2z^3$.To do:We have to find the greatest common factor of the given polynomials.Solution:HCF:A common factor of two or more numbers is a factor that is shared by the numbers. The highest common factor (HCF) of those numbers is found by finding all common factors of the numbers and selecting the largest one.The numerical coefficient of $42x^2yz$ is $42$The numerical coefficient of $63x^3y^2z^3$ is $63$This implies, $42=2\times3\times7$$63=3\times3\times7$HCF of $42$ and $63$ is $3\times7=21$The common variables in the given polynomials are $x, y$ and $z$The power of $x$ in $42x^2yz$ is $2$The power of $x$ in $63x^3y^2z^3$ is $3$The power of $y$ ... Read More

Given:Given polynomials are $7x, 21x^2$ and $14xy^2$.To do:We have to find the greatest common factor of the given polynomials.Solution:GCF/HCF:A common factor of two or more numbers is a factor that is shared by the numbers. The greatest/highest common factor (GCF/HCF) of those numbers is found by finding all common factors of the numbers and selecting the largest one.The numerical coefficient of $7x$ is $7$The numerical coefficient of $21x^2$ is $21$The numerical coefficient of $14xy^2$ is $14$This implies, $7=7\times1$$21=3\times7$$14=2\times7$HCF of $7, 21$ and $14$ is $7$The common variables in the given polynomials are $x$ and $y$The power of $x$ in $7x$ is $1$The power of $x$ ... Read More

Given:Given polynomials are $6x^3y$ and $18x^2y^3$.To do:We have to find the greatest common factor of the given polynomials.Solution:GCF:A common factor of two or more numbers is a factor that is shared by the numbers. The greatest common factor (GCF) of those numbers is found by finding all common factors of the numbers and selecting the largest one.The numerical coefficient of $6x^3y$ is $6$The numerical coefficient of $18x^2y^3$ is $18$This implies, $6=2\times3$$18=2\times3\times3$GCF of $6$ and $18$ is $2\times3=6$The common variables in the given polynomials are $x$ and $y$The power of $x$ in $6x^3y$ is $3$The power of $x$ in $18x^2y^3$ is $2$The power of $y$ in $6x^3y$ ... Read More

Given:Given polynomials are $2x^2$ and $12x^2$.To do:We have to find the greatest common factor of the given polynomials.Solution:HCF:A common factor of two or more numbers is a factor that is shared by the numbers. The highest common factor (HCF) of those numbers is found by finding all common factors of the numbers and selecting the largest one.The numerical coefficient of $2x^2$ is $2$The numerical coefficient of $12x^2$ is $12$This implies, $2=2\times1$$12=2\times2\times3$HCF of $2$ and $12$ is $2$The common variable in the given polynomials is $x$The power of $x$ in $2x^2$ is $2$The power of $x$ in $12x^2$ is $2$The monomial of common literals with the ... Read More

Given:(i) $102 \times 106$(ii) $109 \times 107$(iii) $35 \times 37$(iv) $53 \times 55$(v) $103 \times 96$(vi) $34 \times 36$(vii) $994 \times 1006$To do:We have to find the given products.Solution:Here, to find the given products we can use distributive property twice.Distributive Property:The distributive property of multiplication states that when a factor is multiplied by the sum or difference of two terms, it is essential to multiply each of the two numbers by the factor, and finally perform the addition or subtraction operation.$(a+b)(c+d)=a(c+d)+b(c+d)$..............(I)(i) The given expression is $102 \times 106$We can write $102$ as $102=100+2$ and $106$ as $106=100+6$Therefore, $102 \times 106=(100+2)\times(100+6)$$102 \times ... Read More

To do:We have to find the given products.Solution:Here, to find the given products we can use distributive property twice.Distributive Property:The distributive property of multiplication states that when a factor is multiplied by the sum or difference of two terms, it is essential to multiply each of the two numbers by the factor, and finally perform the addition or subtraction operation.$(a+b)(c+d)=a(c+d)+b(c+d)$..............(I)Therefore, (i) The given expression is $(x + 4) (x + 7)$.$(x + 4) (x + 7)=x(x+7)+4(x+7)$                      [Using (I)]$(x + 4) (x + 7)=x(x)+x(7)+4(x)+4(7)$$(x + 4) (x + 7)=x^2+7x+4x+28$$(x + 4) (x ... Read More

To do:We have to show that:(i) $(3x + 7)^2 - 84x = (3x - 7)^2$(ii) $(9a - 5b)^2 + 180ab = (9a + 5b)^2$(iii) $(\frac{4m}{3} - \frac{3n}{4})^2 + 2mn = \frac{16m^2}{9} + \frac{9n^2}{16}$(iv) $(4pq + 3q)^2 - (4pq - 3q)^2 = 48pq^2$(v) $(a - b) (a + b) + (b - c) (b + c) + (c - a) (c + a) = 0$Solution:To show that LHS $=$ RHS in each case, we can use the following algebraic identities:$(a+b)^2=a^2+2ab+b^2$.............(I)$(a-b)^2=a^2-2ab+b^2$.............(II)$(a+b)(a-b)=a^2-b^2$................(III)(i)  The given equation is $(3x + 7)^2 - 84x = (3x - 7)^2$Let us consider LHS, $(3x + 7)^2 - 84x ... Read More

Given:(i) $(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)$(ii) $(2x - 1) (2x + 1) (4x^2 + 1) (16x^4 + 1)$(iii) $(7m - 8n)^2 + (7m + 8n)^2$(iv) $(2.5p - 1.5q)^2 - (1.5p - 2.5q)^2$(v) $(m^2 - n^2m)^2 + 2m^3n^2$To do:We have to simplify the given expressions.Solution:Here, we have to simplify the given expressions. By using the algebraic identities $(a+b)^2=a^2+2ab+b^2$,  $(a-b)^2=a^2-2ab+b^2$ and  $(a+b)(a-b)=a^2-b^2$, we can reduce the given expressions and simplify them. $(a+b)^2=a^2+2ab+b^2$.............(I)$(a-b)^2=a^2-2ab+b^2$.............(II)$(a+b)(a-b)=a^2-b^2$................(III)(i) The given expression is $(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)$.$(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)=(x^2-y^2)(x^2+y^2)(x^4+y^4)$  ... Read More

Given:The given expressions are(i) $4x^2 - 12x + 7$(ii) $4x^2 - 20x + 20$To do:We have to find the term that must be added to each of the given expression to make it a whole square.Solution:The given expressions are (i) $4x^2 - 12x + 7$ (ii) $4x^2 - 20x + 20$. Here, we have to find the term that must be added to each of the given expression to make it a whole square. So, to find the term that must be added, we have to make the given expressions as the sum of a whole square and some other term and using the ... Read More

Given:$x^2 + y^2 = 29$ and $xy = 2$To do:We have to find the value of(i) $x + y$(ii) $x - y$(iii) $x^4 + y^4$Solution:The given expressions are $x^2 + y^2 = 29$ and $xy = 2$. Here, we have to find the value of (i) $x + y$ (ii) $x - y$ (iii) $x^4 + y^4$. So, by using the identities $(a+b)^2=a^2+2ab+b^2$ and $(a-b)^2=a^2-2ab+b^2$, we can find the required values.$xy = 2$.........(I)$(a+b)^2=a^2+2ab+b^2$.............(II)$(a-b)^2=a^2-2ab+b^2$.............(III)(i) Let us consider, $x^2 + y^2 = 29$Adding $2xy$ on both sides, we get, $x^2+2xy+y^2=29+2xy$$(x+y)^2=29+2(2)$                    [Using (II) and (I)]$(x+y)^2=29+4$$(x+y)^2=33$Taking square root on both sides, ... Read More