Algebraic Identities Questions in English

(A) To evaluate $93 \times 95$ without direct multiplication,we can use the algebraic identity $(x + a)(x + b) = x^2 + (a + b)x + ab$.
We can express the numbers as:
$93 = 90 + 3$
$95 = 90 + 5$
Now,substitute these into the identity where $x = 90$,$a = 3$,and $b = 5$:
$(90 + 3)(90 + 5) = (90)^2 + (3 + 5)(90) + (3 \times 5)$
Calculate each term:
$(90)^2 = 8100$
$(3 + 5)(90) = 8 \times 90 = 720$
$(3 \times 5) = 15$
Adding these values together:
$8100 + 720 + 15 = 8835$
Therefore,$93 \times 95 = 8835$.

(B) To evaluate $78 \times 84$ using algebraic identities,we can express the numbers as:
$78 = (80 - 2)$
$84 = (80 + 4)$
Using the identity $(x + a)(x + b) = x^2 + (a + b)x + ab$,where $x = 80$,$a = -2$,and $b = 4$:
$(80 - 2)(80 + 4) = (80)^2 + (-2 + 4)(80) + (-2)(4)$
$= 6400 + (2)(80) - 8$
$= 6400 + 160 - 8$
$= 6560 - 8$
$= 6552$

(A) To expand the expression $(2 x+5 y)^{2}$,we use the algebraic identity $(a+b)^{2} = a^{2}+2 a b+b^{2}$.
Here,$a = 2 x$ and $b = 5 y$.
Substituting these values into the identity:
$(2 x+5 y)^{2} = (2 x)^{2} + 2(2 x)(5 y) + (5 y)^{2}$
$= 4 x^{2} + 20 x y + 25 y^{2}$.

(A) To expand the expression $(3a - 4)^2$,we use the algebraic identity $(x - y)^2 = x^2 - 2xy + y^2$.
Here,$x = 3a$ and $y = 4$.
Substituting these values into the identity:
$(3a - 4)^2 = (3a)^2 - 2(3a)(4) + (4)^2$
$= 9a^2 - 24a + 16$.

(A) The given expression is $16 x^{2}+40 x y+25 y^{2}$.
We can rewrite this expression in the form of the algebraic identity $(a+b)^{2} = a^{2}+2ab+b^{2}$.
Here,$a^{2} = 16 x^{2} = (4 x)^{2}$,so $a = 4 x$.
And $b^{2} = 25 y^{2} = (5 y)^{2}$,so $b = 5 y$.
Now,check the middle term: $2ab = 2(4 x)(5 y) = 40 x y$.
Since the expression matches the identity,we can write it as $(4 x+5 y)^{2}$.
Thus,the factors are $(4 x+5 y)(4 x+5 y)$.

(B) To factorise the expression $49 x^{2}-42 x+9$,we observe that it follows the algebraic identity $a^{2}-2ab+b^{2} = (a-b)^{2}$.
Here,$a^{2} = 49 x^{2} = (7 x)^{2}$,so $a = 7 x$.
And $b^{2} = 9 = (3)^{2}$,so $b = 3$.
Now,check the middle term: $-2ab = -2(7 x)(3) = -42 x$.
Since the expression matches the identity,we can write:
$49 x^{2}-42 x+9 = (7 x)^{2}-2(7 x)(3)+(3)^{2} = (7 x-3)^{2}$.
Thus,the factors are $(7 x-3)(7 x-3)$.

(C) To evaluate $(107)^{2}$,we can express $107$ as $(100 + 7)$.
Using the algebraic identity $(a + b)^{2} = a^{2} + 2ab + b^{2}$,where $a = 100$ and $b = 7$:
$(100 + 7)^{2} = (100)^{2} + 2(100)(7) + (7)^{2}$
$= 10000 + 1400 + 49$
$= 11449$

(A) The given expression is in the form of the algebraic identity $(a + b)(a - b) = a^2 - b^2$.
Here,$a = 3x$ and $b = 7y$.
Applying the identity:
$(3x + 7y)(3x - 7y) = (3x)^2 - (7y)^2$
$= 9x^2 - 49y^2$

(A) The given expression is $121 x^{2}-289 y^{2}$.
We know that $121 = 11^{2}$ and $289 = 17^{2}$.
So,the expression can be written as $(11 x)^{2}-(17 y)^{2}$.
Using the algebraic identity $a^{2}-b^{2}=(a-b)(a+b)$,where $a=11x$ and $b=17y$,we get:
$(11 x)^{2}-(17 y)^{2}=(11 x-17 y)(11 x+17 y)$.

(A) To evaluate $66 \times 74$ without direct multiplication,we can express the numbers as $(70 - 4)$ and $(70 + 4)$.
Using the algebraic identity $(a - b)(a + b) = a^2 - b^2$,where $a = 70$ and $b = 4$:
$66 \times 74 = (70 - 4)(70 + 4)$
$= (70)^2 - (4)^2$
$= 4900 - 16$
$= 4884$

(A) To expand $(x+3y-5z)^{2}$,we use the algebraic identity: $(a+b+c)^{2} = a^{2}+b^{2}+c^{2}+2ab+2bc+2ca$.
Here,$a = x$,$b = 3y$,and $c = -5z$.
Substituting these values into the identity:
$(x+3y-5z)^{2} = (x)^{2} + (3y)^{2} + (-5z)^{2} + 2(x)(3y) + 2(3y)(-5z) + 2(-5z)(x)$
$= x^{2} + 9y^{2} + 25z^{2} + 6xy - 30yz - 10zx$.

(A) To expand $(2x - y - 5)^2$,we use the algebraic identity: $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$.
Here,$a = 2x$,$b = -y$,and $c = -5$.
Substituting these values into the identity:
$(2x - y - 5)^2 = (2x)^2 + (-y)^2 + (-5)^2 + 2(2x)(-y) + 2(-y)(-5) + 2(-5)(2x)$.
Calculating each term:
$= 4x^2 + y^2 + 25 - 4xy + 10y - 20x$.

(A) We use the algebraic identity: $(a+b+c)^{2} = a^{2}+b^{2}+c^{2}+2ab+2bc+2ca$.
Given expression: $x^{2}+9 y^{2}+4+6 x y+12 y+4 x$.
We can rewrite this as:
$= (x)^{2} + (3y)^{2} + (2)^{2} + 2(x)(3y) + 2(3y)(2) + 2(2)(x)$.
Comparing this with the identity,where $a=x$,$b=3y$,and $c=2$,we get:
$= (x+3y+2)^{2}$.
Thus,the factors are $(x+3y+2)(x+3y+2)$.

(A) The given expression is $x^{2}+4 y^{2}+9 z^{2}-4 x y-12 y z+6 z x$.
We use the algebraic identity: $(a+b+c)^{2} = a^{2}+b^{2}+c^{2}+2ab+2bc+2ca$.
Comparing the given expression with the identity,we can rewrite it as:
$x^{2}+(-2 y)^{2}+(3 z)^{2}+2(x)(-2 y)+2(-2 y)(3 z)+2(3 z)(x)$.
Here,$a = x$,$b = -2y$,and $c = 3z$.
Thus,the expression simplifies to $(x-2y+3z)^{2}$.
Therefore,the factors are $(x-2y+3z)(x-2y+3z)$.

(C) To evaluate $(132)^{2}$ using algebraic identities,we can express $132$ as $(130 + 2)$.
Using the identity $(a + b)^{2} = a^{2} + 2ab + b^{2}$,where $a = 130$ and $b = 2$:
$(130 + 2)^{2} = (130)^{2} + 2(130)(2) + (2)^{2}$
$= 16900 + 520 + 4$
$= 17424$.

To expand $(3x + 2y)^3$,we use the algebraic identity: $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$.
Here,$a = 3x$ and $b = 2y$.
Substituting these values into the identity:
$(3x + 2y)^3 = (3x)^3 + (2y)^3 + 3(3x)(2y)(3x + 2y)$
$= 27x^3 + 8y^3 + 18xy(3x + 2y)$
$= 27x^3 + 8y^3 + 54x^2y + 36xy^2$

(N/A) To expand the expression $(2a - 5b)^3$,we use the algebraic identity:
$(x - y)^3 = x^3 - y^3 - 3xy(x - y)$
Here,$x = 2a$ and $y = 5b$.
Substituting these values into the identity:
$= (2a)^3 - (5b)^3 - 3(2a)(5b)(2a - 5b)$
$= 8a^3 - 125b^3 - 30ab(2a - 5b)$
$= 8a^3 - 125b^3 - 60a^2b + 150ab^2$

(N/A) The given expression is $8 x^{3}+27 y^{3}+36 x^{2} y+54 x y^{2}$.
We can rewrite this expression using the algebraic identity $(a+b)^{3} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2}$.
Here,let $a = 2x$ and $b = 3y$.
Then,$a^{3} = (2x)^{3} = 8x^{3}$ and $b^{3} = (3y)^{3} = 27y^{3}$.
The middle terms are $3a^{2}b = 3(2x)^{2}(3y) = 3(4x^{2})(3y) = 36x^{2}y$ and $3ab^{2} = 3(2x)(3y)^{2} = 3(2x)(9y^{2}) = 54xy^{2}$.
Thus,the expression is of the form $(2x)^{3} + (3y)^{3} + 3(2x)^{2}(3y) + 3(2x)(3y)^{2}$.
This simplifies to $(2x + 3y)^{3}$.
Therefore,the factorised form is $(2x + 3y)(2x + 3y)(2x + 3y)$.

(N/A) The given expression is $27 x^{3}-64-108 x^{2}+144 x$.
We can rewrite this expression in the form $a^{3} + b^{3} + 3a^{2}b + 3ab^{2} = (a + b)^{3}$.
Here,$a = 3x$ and $b = -4$.
Substituting these values into the identity:
$27 x^{3}-64-108 x^{2}+144 x = (3x)^{3} + (-4)^{3} + 3(3x)^{2}(-4) + 3(3x)(-4)^{2}$.
This simplifies to $(3x - 4)^{3}$.
Thus,the factorised form is $(3x - 4)(3x - 4)(3x - 4)$.

(D) To evaluate $(998)^{3}$,we can express $998$ as $(1000 - 2)$.
Using the algebraic identity $(a - b)^{3} = a^{3} - b^{3} - 3ab(a - b)$,where $a = 1000$ and $b = 2$:
$(1000 - 2)^{3} = (1000)^{3} - (2)^{3} - 3(1000)(2)(1000 - 2)$
$= 1,000,000,000 - 8 - 6000(998)$
$= 1,000,000,000 - 8 - 5,988,000$
$= 994,011,992$

(D) The given expression is $8 x^{3}+y^{3}-27 z^{3}+18 x y z$.
We use the algebraic identity: $a^{3}+b^{3}+c^{3}-3 a b c = (a+b+c)(a^{2}+b^{2}+c^{2}-a b-b c-c a)$.
Rewrite the expression as: $(2 x)^{3}+(y)^{3}+(-3 z)^{3}-3(2 x)(y)(-3 z)$.
Here,$a = 2 x$,$b = y$,and $c = -3 z$.
Applying the identity:
$= (2 x+y-3 z)((2 x)^{2}+(y)^{2}+(-3 z)^{2}-(2 x)(y)-(y)(-3 z)-(-3 z)(2 x))$.
Simplifying the terms:
$= (2 x+y-3 z)(4 x^{2}+y^{2}+9 z^{2}-2 x y+3 y z+6 z x)$.

(B) Let $a = 21$,$b = 15$,and $c = -36$.
First,calculate the sum of $a, b,$ and $c$:
$a + b + c = 21 + 15 + (-36) = 36 - 36 = 0$.
We know the algebraic identity: If $a + b + c = 0$,then $a^{3} + b^{3} + c^{3} = 3abc$.
Substituting the values into the identity:
$(21)^{3} + (15)^{3} + (-36)^{3} = 3 \times (21) \times (15) \times (-36)$.
Calculating the product:
$3 \times 21 = 63$
$15 \times (-36) = -540$
$63 \times (-540) = -34020$.
Therefore,the value is $-34020$.

(B) To expand the expression $(x+4)(x+9)$,we use the distributive property ($FOIL$ method):
$(x+4)(x+9) = x(x+9) + 4(x+9)$
$= x^{2} + 9x + 4x + 36$
$= x^{2} + 13x + 36$

(A) To expand the expression $(3x - 1)(3x + 4)$,we use the distributive property ($FOIL$ method):
$(3x - 1)(3x + 4) = (3x)(3x) + (3x)(4) + (-1)(3x) + (-1)(4)$
$= 9x^2 + 12x - 3x - 4$
$= 9x^2 + 9x - 4$

(A) To expand the expression $(2x - 7)(2x - 5)$,we use the distributive property ($FOIL$ method):
$1$. Multiply the first terms: $(2x) \times (2x) = 4x^2$.
$2$. Multiply the outer terms: $(2x) \times (-5) = -10x$.
$3$. Multiply the inner terms: $(-7) \times (2x) = -14x$.
$4$. Multiply the last terms: $(-7) \times (-5) = 35$.
Combining these results: $4x^2 - 10x - 14x + 35 = 4x^2 - 24x + 35$.

(N/A) To expand the expression $(2a + 3b)(2a - 5b)$,we use the distributive property ($FOIL$ method):
$1$. Multiply the first terms: $(2a) \times (2a) = 4a^2$.
$2$. Multiply the outer terms: $(2a) \times (-5b) = -10ab$.
$3$. Multiply the inner terms: $(3b) \times (2a) = 6ab$.
$4$. Multiply the last terms: $(3b) \times (-5b) = -15b^2$.
Combining these results: $4a^2 - 10ab + 6ab - 15b^2$.
Finally,simplify the middle terms: $-10ab + 6ab = -4ab$.
Thus,the expanded form is $4a^2 - 4ab - 15b^2$.

(C) To evaluate $103 \times 105$ without direct multiplication,we can use the algebraic identity: $(x + a)(x + b) = x^2 + (a + b)x + ab$.
Here,we can write $103 = (100 + 3)$ and $105 = (100 + 5)$.
Comparing this with the identity,we have $x = 100$,$a = 3$,and $b = 5$.
Substituting these values into the identity:
$(100 + 3)(100 + 5) = (100)^2 + (3 + 5)(100) + (3 \times 5)$
$= 10000 + (8 \times 100) + 15$
$= 10000 + 800 + 15$
$= 10815$.

(D) To evaluate $84 \times 79$ without direct multiplication,we can express the numbers as binomials.
We can write $84$ as $(80 + 4)$ and $79$ as $(80 - 1)$.
Using the algebraic identity $(x + a)(x + b) = x^2 + (a + b)x + ab$,where $x = 80$,$a = 4$,and $b = -1$:
$(80 + 4)(80 - 1) = 80^2 + (4 - 1)80 + (4 \times -1)$
$= 6400 + (3 \times 80) - 4$
$= 6400 + 240 - 4$
$= 6640 - 4$
$= 6636$.

(A) To evaluate $76 \times 82$ without direct multiplication,we can express the numbers as $(79 - 3)$ and $(79 + 3)$.
Using the algebraic identity $(a - b)(a + b) = a^2 - b^2$,where $a = 79$ and $b = 3$:
$76 \times 82 = (79 - 3)(79 + 3)$
$= 79^2 - 3^2$
$= 6241 - 9$
$= 6232$.

(B) To evaluate $88 \times 86$ without direct multiplication,we can express the numbers as $(90 - 2) \times (90 - 4)$.
Using the algebraic identity $(x + a)(x + b) = x^2 + (a + b)x + ab$,where $x = 90$,$a = -2$,and $b = -4$:
$(90 - 2)(90 - 4) = 90^2 + (-2 - 4) \times 90 + (-2 \times -4)$
$= 8100 + (-6 \times 90) + 8$
$= 8100 - 540 + 8$
$= 7560 + 8$
$= 7568$.

(A) To expand the expression $(3x + 5)^2$,we use the algebraic identity $(a + b)^2 = a^2 + 2ab + b^2$.
Here,$a = 3x$ and $b = 5$.
Substituting these values into the identity:
$(3x + 5)^2 = (3x)^2 + 2(3x)(5) + (5)^2$
$= 9x^2 + 30x + 25$.

(B) To expand the expression $(6x - 7)^2$,we use the algebraic identity $(a - b)^2 = a^2 - 2ab + b^2$.
Here,$a = 6x$ and $b = 7$.
Substituting these values into the identity:
$(6x - 7)^2 = (6x)^2 - 2(6x)(7) + (7)^2$
$= 36x^2 - 84x + 49$.
Thus,the expanded form is $36x^2 - 84x + 49$.

(B) To expand $(2a + 3b)^2$,we use the algebraic identity $(x + y)^2 = x^2 + 2xy + y^2$.
Here,$x = 2a$ and $y = 3b$.
Substituting these values into the identity:
$(2a + 3b)^2 = (2a)^2 + 2(2a)(3b) + (3b)^2$
$= 4a^2 + 12ab + 9b^2$.

(A) To expand the expression $\left(\frac{x}{2}-\frac{2}{5}\right)^{2}$,we use the algebraic identity $(a-b)^{2} = a^{2} - 2ab + b^{2}$.
Here,$a = \frac{x}{2}$ and $b = \frac{2}{5}$.
Substituting these values into the identity:
$\left(\frac{x}{2}-\frac{2}{5}\right)^{2} = \left(\frac{x}{2}\right)^{2} - 2\left(\frac{x}{2}\right)\left(\frac{2}{5}\right) + \left(\frac{2}{5}\right)^{2}$
$= \frac{x^{2}}{4} - \frac{4x}{10} + \frac{4}{25}$
$= \frac{x^{2}}{4} - \frac{2x}{5} + \frac{4}{25}$.

(A) To expand the expression $\left(\frac{2x}{3} + \frac{3y}{4}\right)^{2}$,we use the algebraic identity $(a + b)^{2} = a^{2} + 2ab + b^{2}$.
Here,$a = \frac{2x}{3}$ and $b = \frac{3y}{4}$.
Substituting these values into the identity:
$\left(\frac{2x}{3} + \frac{3y}{4}\right)^{2} = \left(\frac{2x}{3}\right)^{2} + 2\left(\frac{2x}{3}\right)\left(\frac{3y}{4}\right) + \left(\frac{3y}{4}\right)^{2}$
$= \frac{4x^{2}}{9} + 2\left(\frac{6xy}{12}\right) + \frac{9y^{2}}{16}$
$= \frac{4}{9}x^{2} + 2\left(\frac{xy}{2}\right) + \frac{9}{16}y^{2}$
$= \frac{4}{9}x^{2} + xy + \frac{9}{16}y^{2}$.

(A) To expand the expression $(x - \frac{1}{2})^{2}$,we use the algebraic identity $(a - b)^{2} = a^{2} - 2ab + b^{2}$.
Here,$a = x$ and $b = \frac{1}{2}$.
Substituting these values into the identity:
$(x - \frac{1}{2})^{2} = (x)^{2} - 2(x)(\frac{1}{2}) + (\frac{1}{2})^{2}$.
Simplifying the terms:
$= x^{2} - x + \frac{1}{4}$.

(B) The given expression is $16 x^{2}-40 x y+25 y^{2}$.
We can rewrite this expression as $(4 x)^{2}-2(4 x)(5 y)+(5 y)^{2}$.
This is in the form of the algebraic identity $a^{2}-2 a b+b^{2}=(a-b)^{2}$,where $a=4 x$ and $b=5 y$.
Substituting these values,we get $(4 x-5 y)^{2}$.

(A) To factorise the expression $9x^{2} + 42x + 49$,we observe that it follows the algebraic identity $(a + b)^{2} = a^{2} + 2ab + b^{2}$.
We can rewrite the expression as:
$9x^{2} + 42x + 49 = (3x)^{2} + 2(3x)(7) + (7)^{2}$.
Here,$a = 3x$ and $b = 7$.
Therefore,the expression simplifies to $(3x + 7)^{2}$.

(A) The given expression is $\frac{4x^2}{9} - \frac{x}{3} + \frac{1}{16}$.
This can be rewritten in the form $a^2 - 2ab + b^2 = (a - b)^2$.
Here,$a^2 = \frac{4x^2}{9} \implies a = \frac{2x}{3}$.
And $b^2 = \frac{1}{16} \implies b = \frac{1}{4}$.
Checking the middle term: $-2ab = -2 \times \left(\frac{2x}{3}\right) \times \left(\frac{1}{4}\right) = -\frac{4x}{12} = -\frac{x}{3}$.
Since the middle term matches,the expression is $\left(\frac{2x}{3} - \frac{1}{4}\right)^2$.

(A) The given expression is $\frac{x^{2}}{4}+\frac{3 x y}{5}+\frac{9 y^{2}}{25}$.
This can be written in the form $a^{2} + 2ab + b^{2} = (a + b)^{2}$.
Here,$a^{2} = \frac{x^{2}}{4} \implies a = \frac{x}{2}$.
And $b^{2} = \frac{9 y^{2}}{25} \implies b = \frac{3 y}{5}$.
Now,check the middle term $2ab = 2 \times \left(\frac{x}{2}\right) \times \left(\frac{3 y}{5}\right) = \frac{3 x y}{5}$.
Since the middle term matches,the expression is $\left(\frac{x}{2}+\frac{3 y}{5}\right)^{2}$.

(A) To evaluate $(65)^{2}$,we can use the algebraic identity $(a + b)^{2} = a^{2} + 2ab + b^{2}$.
Here,we can write $65$ as $(60 + 5)$.
So,$(65)^{2} = (60 + 5)^{2}$.
Using the identity: $(60)^{2} + 2(60)(5) + (5)^{2}$.
$= 3600 + 600 + 25$.
$= 4225$.
Alternatively,for any number ending in $5$,the square is calculated as $(n5)^{2} = [n(n+1)] \times 100 + 25$.
For $65$,$n = 6$,so $[6(6+1)] \times 100 + 25 = (6 \times 7) \times 100 + 25 = 4200 + 25 = 4225$.

(B) To evaluate $(101)^{2}$,we can express $101$ as $(100 + 1)$.
Using the algebraic identity $(a + b)^{2} = a^{2} + 2ab + b^{2}$,where $a = 100$ and $b = 1$:
$(100 + 1)^{2} = (100)^{2} + 2(100)(1) + (1)^{2}$
$= 10000 + 200 + 1$
$= 10201$.

(C) To evaluate $(1002)^{2}$,we can express $1002$ as $(1000 + 2)$.
Using the algebraic identity $(a + b)^{2} = a^{2} + 2ab + b^{2}$,where $a = 1000$ and $b = 2$:
$(1000 + 2)^{2} = (1000)^{2} + 2(1000)(2) + (2)^{2}$
$= 1000000 + 4000 + 4$
$= 1004004$.

(D) To evaluate $(98)^{2}$,we can express $98$ as $(100 - 2)$.
Using the algebraic identity $(a - b)^{2} = a^{2} - 2ab + b^{2}$,where $a = 100$ and $b = 2$:
$(100 - 2)^{2} = (100)^{2} - 2(100)(2) + (2)^{2}$
$= 10000 - 400 + 4$
$= 9600 + 4$
$= 9604$.

(A) To expand the expression $(x+5y)(x-5y)$,we use the algebraic identity $(a+b)(a-b) = a^2 - b^2$.
Here,$a = x$ and $b = 5y$.
Substituting these values into the identity:
$(x+5y)(x-5y) = (x)^2 - (5y)^2$
$= x^2 - 25y^2$.

(B) The given expression is in the form of $(a + b)(a - b)$,where $a = \frac{2x}{3}$ and $b = \frac{4y}{5}$.
Using the algebraic identity $(a + b)(a - b) = a^2 - b^2$,we get:
$\left(\frac{2x}{3}\right)^2 - \left(\frac{4y}{5}\right)^2$
$= \frac{4x^2}{9} - \frac{16y^2}{25}$
$= \frac{4}{9}x^2 - \frac{16}{25}y^2$.

(N/A) To expand the expression $(3x + 7)(-3x + 7)$,we can rearrange the terms to use the algebraic identity $(a + b)(a - b) = a^2 - b^2$.
First,rewrite the expression as $(7 + 3x)(7 - 3x)$.
Here,$a = 7$ and $b = 3x$.
Applying the identity: $(7)^2 - (3x)^2 = 49 - 9x^2$.

(A) To expand the expression $(11 x+18)(11 x-18)$,we use the algebraic identity $(a+b)(a-b) = a^{2}-b^{2}$.
Here,$a = 11 x$ and $b = 18$.
Applying the identity:
$(11 x)^{2} - (18)^{2}$
$= 121 x^{2} - 324$.

(A) The given expression is $\frac{4x^2}{9} - \frac{1}{25}$.
This can be written in the form of $a^2 - b^2$,where $a^2 = \frac{4x^2}{9}$ and $b^2 = \frac{1}{25}$.
Taking the square root,we get $a = \frac{2x}{3}$ and $b = \frac{1}{5}$.
Using the algebraic identity $a^2 - b^2 = (a + b)(a - b)$,we substitute the values of $a$ and $b$.
Therefore,$\frac{4x^2}{9} - \frac{1}{25} = \left(\frac{2x}{3} + \frac{1}{5}\right)\left(\frac{2x}{3} - \frac{1}{5}\right)$.

(A) The given expression is $169 x^{2}-625$.
We can write this as $(13 x)^{2}-(25)^{2}$.
Using the algebraic identity $a^{2}-b^{2} = (a+b)(a-b)$,where $a = 13 x$ and $b = 25$,we get:
$(13 x)^{2}-(25)^{2} = (13 x+25)(13 x-25)$.