Algebraic Expressions and Identities - Exercise 6.4 | Mathematics (Maths) Class 8 PDF Download

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Q u e s t i o n : 4 2
Evaluate each of the following when x = 2, y = -1.
35x2y×-154xy2×79x2y2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices, i.e., ? am×an=am+n.
We have:
35x2y×-154xy2×79x2y2=35×-154×79×x2×x×x2×y×y2×y2=35×-154×79×x2+1+2×y1+2+2=-74x5y5
? 35x2y×-154xy2×79x2y2=-74x5y5.
Substituting x = 2 and y = -1 in the result, we get:
-74x5y5=-7425-15=-74×32×-1=56
Thus, the answer is 56.
Q u e s t i o n : 4 3
Find the following product:
2a
3
(3a + 5b)
S o l u t i o n :
To find the product, we will use distributive law as follows:
2a33a+5b=2a3×3a+2a3×5b=2×3a3×a+2×5a3b=2×3a3+1+2×5a3b=6a4+10a3b
Thus, the answer is 6a4+10a3b.
Q u e s t i o n : 4 4
Find the following product:
-11a(3a + 2b)
S o l u t i o n :
To find the product, we will use distributive law as follows:
-11a3a+2b=-11a×3a+-11a×2b=-11×3×a×a+-11×2×a×b=-33×a1+1+-22×a×b=-33a2-22ab
Thus, the answer is -33a2-22ab.
Q u e s t i o n : 4 5
Find the following product:
-5a(7a - 2b)
S o l u t i o n :
To find the product, we will use distributive law as follows:
-5a7a-2b=-5a×7a+-5a×-2b=-5×7×a×a+-5×-2×a×b=-35×a1+1+10×a×b=-35a2+10ab
Thus, the answer is -35a2+10ab.
Q u e s t i o n : 4 6
Find the following product:
-11y
2
3y + 7
S o l u t i o n :
To find the product, we will use distributive law as follows:
-11y23y+7=-11y2×3y+-11y2×7=-11×3y2×y+-11×7×y2=-33y2+1+-77×y2=-33y3-77y2
Thus, the answer is -33y3-77y2.
Q u e s t i o n : 4 7
Find the following product:
6x5(x3+y3)
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Q u e s t i o n : 4 2
Evaluate each of the following when x = 2, y = -1.
35x2y×-154xy2×79x2y2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices, i.e., ? am×an=am+n.
We have:
35x2y×-154xy2×79x2y2=35×-154×79×x2×x×x2×y×y2×y2=35×-154×79×x2+1+2×y1+2+2=-74x5y5
? 35x2y×-154xy2×79x2y2=-74x5y5.
Substituting x = 2 and y = -1 in the result, we get:
-74x5y5=-7425-15=-74×32×-1=56
Thus, the answer is 56.
Q u e s t i o n : 4 3
Find the following product:
2a
3
(3a + 5b)
S o l u t i o n :
To find the product, we will use distributive law as follows:
2a33a+5b=2a3×3a+2a3×5b=2×3a3×a+2×5a3b=2×3a3+1+2×5a3b=6a4+10a3b
Thus, the answer is 6a4+10a3b.
Q u e s t i o n : 4 4
Find the following product:
-11a(3a + 2b)
S o l u t i o n :
To find the product, we will use distributive law as follows:
-11a3a+2b=-11a×3a+-11a×2b=-11×3×a×a+-11×2×a×b=-33×a1+1+-22×a×b=-33a2-22ab
Thus, the answer is -33a2-22ab.
Q u e s t i o n : 4 5
Find the following product:
-5a(7a - 2b)
S o l u t i o n :
To find the product, we will use distributive law as follows:
-5a7a-2b=-5a×7a+-5a×-2b=-5×7×a×a+-5×-2×a×b=-35×a1+1+10×a×b=-35a2+10ab
Thus, the answer is -35a2+10ab.
Q u e s t i o n : 4 6
Find the following product:
-11y
2
3y + 7
S o l u t i o n :
To find the product, we will use distributive law as follows:
-11y23y+7=-11y2×3y+-11y2×7=-11×3y2×y+-11×7×y2=-33y2+1+-77×y2=-33y3-77y2
Thus, the answer is -33y3-77y2.
Q u e s t i o n : 4 7
Find the following product:
6x5(x3+y3)
S o l u t i o n :
To find the product, we will use distributive law as follows:
6x5x3+y3=6x5×x3+6x5×y3=65×x×x3+65×x×y3=65×x1+3+65×x×y3=6x45+6xy35
Thus, the answer is 6x45+6xy35.
Q u e s t i o n : 4 8
xy(x
3
 - y
3
)
S o l u t i o n :
To find the product, we will use the distributive law in the following way:
xyx3-y3=xy×x3-xy×y3=x×x3×y-x×y×y3=x1+3y-xy1+3=x4y-xy4
Thus, the answer is x4y-xy4.
Q u e s t i o n : 4 9
Find the following product:
0.1y(0.1x
5
 + 0.1y)
S o l u t i o n :
To find the product, we will use distributive law as follows:
0.1y0.1x5+0.1y=0.1y0.1x5+0.1y0.1y=0.1×0.1y×x5+0.1×0.1y×y=0.1×0.1x5×y+0.1×0.1y1+1=0.01x5y+0.01y2
Thus, the answer is 0.01x5y+0.01y2.
Q u e s t i o n : 5 0
Find the following product:
-74ab2c-625a2c2(-50a2b2c2)
S o l u t i o n :
To find the product, we will use distributive law as follows:
-74ab2c-625a2c2-50a2b2c2=-74ab2c-50a2b2c2-625a2c2-50a2b2c2=-74×-50a×a2×b2×b2×c×c2-625-50a2×a2×b2×c2×c2=-74×-50a1+2b2+2c1+2-625-50a2+2b2c2+2=1752a3b4c3--12a4b2
=1752a3b4c3+12a4b2c4
Thus, the answer is 1752a3b4c3+12a4b2c4.
Q u e s t i o n : 5 1
Find the following product:
-827xyz32xyz2-94xy2z3
S o l u t i o n :
To find the product, we will use the distributive law in the following way:
-827xyz32xyz2-94xy2z3=-827xyz32xyz2--827xyz94xy2z3=-827×32x×x×y×y×z×z2--827×94x×x×y×y2×z×z3=-827×32x1+1y1+1z1+2--827×94x1+1y1+2z1+3=-84279×32x1+1y1+1z1+2--82
=-49x2y2z3+23x2y3z4
Thus, the answer is -49x2y2z3+23x2y3z4.
Q u e s t i o n : 5 2
Find the following product:
-427xyz92x2yz-34xyz2
S o l u t i o n :
To find the product, we will use distributive law as follows:
-427xyz92x2yz-34xyz2=-427xyz92x2yz--427xyz34xyz2=-427×92x1+2y1+1z1+1--427×34x1+1y1+1z1+2=-42273×92x1+2y1+1z1+1--41279×34x1+1y1+1z1+2=-23x3y2z2+19x2y2z3
Thus, the answer is -23x3y2z2+19x2y2z3.
Q u e s t i o n : 5 3
Find the following product:
1.5x(10x
2
y - 100xy
2
)
S o l u t i o n :
To find the product, we will use distributive law as follows:
1.5x10x2y-100xy2=1.5x×10x2y-1.5x×100xy2=15x1+2y-150x1+1y2=15x3y-150x2y2
Thus, the answer is 15x3y-150x2y2.
Q u e s t i o n : 5 4
Find the following product:
4.1xy(1.1x - y)
S o l u t i o n :
To find the product, we will use distributive law as follows:
4.1xy1.1x-y=4.1xy×1.1x-4.1xy×y=4.1×1.1×xy×x-4.1xy×y=4.51x1+1y-4.1xy1+1=4.51x2y-4.1xy2
Thus, the answer is 4.51x2y-4.1xy2.
Q u e s t i o n : 5 5
Find the following product:
250.5xyxz+y10
S o l u t i o n :
To find the product, we will use distributive law as follows:
250.5xyxz+y10=250.5xy×xz+250.5xy×y10=250.5x1+1yz+25.05xy1+1=250.5x2yz+25.05xy2
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Q u e s t i o n : 4 2
Evaluate each of the following when x = 2, y = -1.
35x2y×-154xy2×79x2y2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices, i.e., ? am×an=am+n.
We have:
35x2y×-154xy2×79x2y2=35×-154×79×x2×x×x2×y×y2×y2=35×-154×79×x2+1+2×y1+2+2=-74x5y5
? 35x2y×-154xy2×79x2y2=-74x5y5.
Substituting x = 2 and y = -1 in the result, we get:
-74x5y5=-7425-15=-74×32×-1=56
Thus, the answer is 56.
Q u e s t i o n : 4 3
Find the following product:
2a
3
(3a + 5b)
S o l u t i o n :
To find the product, we will use distributive law as follows:
2a33a+5b=2a3×3a+2a3×5b=2×3a3×a+2×5a3b=2×3a3+1+2×5a3b=6a4+10a3b
Thus, the answer is 6a4+10a3b.
Q u e s t i o n : 4 4
Find the following product:
-11a(3a + 2b)
S o l u t i o n :
To find the product, we will use distributive law as follows:
-11a3a+2b=-11a×3a+-11a×2b=-11×3×a×a+-11×2×a×b=-33×a1+1+-22×a×b=-33a2-22ab
Thus, the answer is -33a2-22ab.
Q u e s t i o n : 4 5
Find the following product:
-5a(7a - 2b)
S o l u t i o n :
To find the product, we will use distributive law as follows:
-5a7a-2b=-5a×7a+-5a×-2b=-5×7×a×a+-5×-2×a×b=-35×a1+1+10×a×b=-35a2+10ab
Thus, the answer is -35a2+10ab.
Q u e s t i o n : 4 6
Find the following product:
-11y
2
3y + 7
S o l u t i o n :
To find the product, we will use distributive law as follows:
-11y23y+7=-11y2×3y+-11y2×7=-11×3y2×y+-11×7×y2=-33y2+1+-77×y2=-33y3-77y2
Thus, the answer is -33y3-77y2.
Q u e s t i o n : 4 7
Find the following product:
6x5(x3+y3)
S o l u t i o n :
To find the product, we will use distributive law as follows:
6x5x3+y3=6x5×x3+6x5×y3=65×x×x3+65×x×y3=65×x1+3+65×x×y3=6x45+6xy35
Thus, the answer is 6x45+6xy35.
Q u e s t i o n : 4 8
xy(x
3
 - y
3
)
S o l u t i o n :
To find the product, we will use the distributive law in the following way:
xyx3-y3=xy×x3-xy×y3=x×x3×y-x×y×y3=x1+3y-xy1+3=x4y-xy4
Thus, the answer is x4y-xy4.
Q u e s t i o n : 4 9
Find the following product:
0.1y(0.1x
5
 + 0.1y)
S o l u t i o n :
To find the product, we will use distributive law as follows:
0.1y0.1x5+0.1y=0.1y0.1x5+0.1y0.1y=0.1×0.1y×x5+0.1×0.1y×y=0.1×0.1x5×y+0.1×0.1y1+1=0.01x5y+0.01y2
Thus, the answer is 0.01x5y+0.01y2.
Q u e s t i o n : 5 0
Find the following product:
-74ab2c-625a2c2(-50a2b2c2)
S o l u t i o n :
To find the product, we will use distributive law as follows:
-74ab2c-625a2c2-50a2b2c2=-74ab2c-50a2b2c2-625a2c2-50a2b2c2=-74×-50a×a2×b2×b2×c×c2-625-50a2×a2×b2×c2×c2=-74×-50a1+2b2+2c1+2-625-50a2+2b2c2+2=1752a3b4c3--12a4b2
=1752a3b4c3+12a4b2c4
Thus, the answer is 1752a3b4c3+12a4b2c4.
Q u e s t i o n : 5 1
Find the following product:
-827xyz32xyz2-94xy2z3
S o l u t i o n :
To find the product, we will use the distributive law in the following way:
-827xyz32xyz2-94xy2z3=-827xyz32xyz2--827xyz94xy2z3=-827×32x×x×y×y×z×z2--827×94x×x×y×y2×z×z3=-827×32x1+1y1+1z1+2--827×94x1+1y1+2z1+3=-84279×32x1+1y1+1z1+2--82
=-49x2y2z3+23x2y3z4
Thus, the answer is -49x2y2z3+23x2y3z4.
Q u e s t i o n : 5 2
Find the following product:
-427xyz92x2yz-34xyz2
S o l u t i o n :
To find the product, we will use distributive law as follows:
-427xyz92x2yz-34xyz2=-427xyz92x2yz--427xyz34xyz2=-427×92x1+2y1+1z1+1--427×34x1+1y1+1z1+2=-42273×92x1+2y1+1z1+1--41279×34x1+1y1+1z1+2=-23x3y2z2+19x2y2z3
Thus, the answer is -23x3y2z2+19x2y2z3.
Q u e s t i o n : 5 3
Find the following product:
1.5x(10x
2
y - 100xy
2
)
S o l u t i o n :
To find the product, we will use distributive law as follows:
1.5x10x2y-100xy2=1.5x×10x2y-1.5x×100xy2=15x1+2y-150x1+1y2=15x3y-150x2y2
Thus, the answer is 15x3y-150x2y2.
Q u e s t i o n : 5 4
Find the following product:
4.1xy(1.1x - y)
S o l u t i o n :
To find the product, we will use distributive law as follows:
4.1xy1.1x-y=4.1xy×1.1x-4.1xy×y=4.1×1.1×xy×x-4.1xy×y=4.51x1+1y-4.1xy1+1=4.51x2y-4.1xy2
Thus, the answer is 4.51x2y-4.1xy2.
Q u e s t i o n : 5 5
Find the following product:
250.5xyxz+y10
S o l u t i o n :
To find the product, we will use distributive law as follows:
250.5xyxz+y10=250.5xy×xz+250.5xy×y10=250.5x1+1yz+25.05xy1+1=250.5x2yz+25.05xy2
Thus, the answer is 250.5x2yz+25.05xy2.
Q u e s t i o n : 5 6
Find the following product:
75x2y35xy2+25x
S o l u t i o n :
To find the product, we will use distributive law as follows:
75x2y35xy2+25x=75x2y×35xy2+75x2y×25x=2125x2+1y1+2+1425x2+1y=2125x3y3+1425x3y
Thus, the answer is 2125x3y3+1425x3y.
Q u e s t i o n : 5 7
Find the following product:
43a(a2 + b2 - 3c2)
S o l u t i o n :
To find the product, we will use distributive law as follows:
43aa2+b2-3c2=43a×a2+43a×b2-43a×3c2=43a1+2+43ab2-4ac2=43a3+43ab2-4ac2
Thus, the answer is 43a3+43ab2-4ac2.
Q u e s t i o n : 5 8
Find the product 24x
2
 (1 - 2x) and evaluate its value for x = 3.
S o l u t i o n :
To find the product, we will use distributive law as follows:
24x21-2x=24x2×1-24x2×2x=24x2-48x1+2=24x2-48x3
Substituting  x = 3 in the result, we get:
24x2-48x3=2432-4833=24×9-48×27=216-1296=-1080
Thus, the product is (24x2-48x3) and its value for x = 3 is (-1080).
Q u e s t i o n : 5 9
Find the product -3y(xy + y
2
) and find its value for x = 4 and y = 5.
S o l u t i o n :
To find the product, we will use distributive law as follows: ? -3yxy+y2=-3y×xy+-3y×y2=-3xy1+1-3y1+2=-3xy2-3y3
Substituting x = 4 and y = 5 in the result, we get:
-3xy2-3y3=-3452-353=-3425-3125=-300-375=-675
Thus, the product is (-3xy2-3y3), and its value for ?x = 4 and y = 5 is (-675).
Q u e s t i o n : 6 0
Multiply -32x2y3 by (2x - y) and verify the answer for x = 1 and y = 2.
S o l u t i o n :
To find the product, we will use distributive law as follows:
-32x2y3×2x-y=-32x2y3×2x--32x2y3×y=-3x2+1y3--32x2y3+1=-3x3y3+32x2y4
Substituting x = 1 and y = 2 in the result, we get:
-3x3y3+32x2y4=-31323+321224=-3×1×8+32×1×16=-24+24=0
Thus, the product is -3x3y3+32x2y4, and its value for ?x = 1 and y = 2 is 0.
Q u e s t i o n : 6 1
Multiply the monomial by the binomial and find the value of each for x = -1, y = 0.25 and z = 0.05:
i 15y
2
(2 - 3x)
ii -3x(y
2
 + z
2
)
iii z
2
(x - y)
iv xz(x
2
 + y
2
)
S o l u t i o n :
i To find the product, we will use distributive law as follows:
15y22-3x=15y2×2-15y2×3x=30y2-45xy2
Substituting x = -1 and y = 0.25 in the result, we get:
30y2-45xy2=300.252-45-10.252=30×0.0625-45×-1×0.0625=30×0.0625-45×-1×0.0625=1.875--2.8125=1.875+2.8125=4.6875
ii To find the product, we will use distributive law as follows:
-3xy2+z2=-3x×y2+-3x×z2=-3xy2-3xz2
Substituting x = -1, y = 0.25 ? and z = 0.05 ? in the result, we get:
-3xy2-3xz2=-3-10.252-3-10.052=-3-10.0625-3-10.0025=01875+0.0075=0.195
iii To find the product, we will use distributive law as follows:
z2x-y=z2×x-z2×y=xz2-yz2
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Q u e s t i o n : 4 2
Evaluate each of the following when x = 2, y = -1.
35x2y×-154xy2×79x2y2
S o l u t i o n :
To multiply algebraic expressions, we use commutative and associative laws along with the laws of indices, i.e., ? am×an=am+n.
We have:
35x2y×-154xy2×79x2y2=35×-154×79×x2×x×x2×y×y2×y2=35×-154×79×x2+1+2×y1+2+2=-74x5y5
? 35x2y×-154xy2×79x2y2=-74x5y5.
Substituting x = 2 and y = -1 in the result, we get:
-74x5y5=-7425-15=-74×32×-1=56
Thus, the answer is 56.
Q u e s t i o n : 4 3
Find the following product:
2a
3
(3a + 5b)
S o l u t i o n :
To find the product, we will use distributive law as follows:
2a33a+5b=2a3×3a+2a3×5b=2×3a3×a+2×5a3b=2×3a3+1+2×5a3b=6a4+10a3b
Thus, the answer is 6a4+10a3b.
Q u e s t i o n : 4 4
Find the following product:
-11a(3a + 2b)
S o l u t i o n :
To find the product, we will use distributive law as follows:
-11a3a+2b=-11a×3a+-11a×2b=-11×3×a×a+-11×2×a×b=-33×a1+1+-22×a×b=-33a2-22ab
Thus, the answer is -33a2-22ab.
Q u e s t i o n : 4 5
Find the following product:
-5a(7a - 2b)
S o l u t i o n :
To find the product, we will use distributive law as follows:
-5a7a-2b=-5a×7a+-5a×-2b=-5×7×a×a+-5×-2×a×b=-35×a1+1+10×a×b=-35a2+10ab
Thus, the answer is -35a2+10ab.
Q u e s t i o n : 4 6
Find the following product:
-11y
2
3y + 7
S o l u t i o n :
To find the product, we will use distributive law as follows:
-11y23y+7=-11y2×3y+-11y2×7=-11×3y2×y+-11×7×y2=-33y2+1+-77×y2=-33y3-77y2
Thus, the answer is -33y3-77y2.
Q u e s t i o n : 4 7
Find the following product:
6x5(x3+y3)
S o l u t i o n :
To find the product, we will use distributive law as follows:
6x5x3+y3=6x5×x3+6x5×y3=65×x×x3+65×x×y3=65×x1+3+65×x×y3=6x45+6xy35
Thus, the answer is 6x45+6xy35.
Q u e s t i o n : 4 8
xy(x
3
 - y
3
)
S o l u t i o n :
To find the product, we will use the distributive law in the following way:
xyx3-y3=xy×x3-xy×y3=x×x3×y-x×y×y3=x1+3y-xy1+3=x4y-xy4
Thus, the answer is x4y-xy4.
Q u e s t i o n : 4 9
Find the following product:
0.1y(0.1x
5
 + 0.1y)
S o l u t i o n :
To find the product, we will use distributive law as follows:
0.1y0.1x5+0.1y=0.1y0.1x5+0.1y0.1y=0.1×0.1y×x5+0.1×0.1y×y=0.1×0.1x5×y+0.1×0.1y1+1=0.01x5y+0.01y2
Thus, the answer is 0.01x5y+0.01y2.
Q u e s t i o n : 5 0
Find the following product:
-74ab2c-625a2c2(-50a2b2c2)
S o l u t i o n :
To find the product, we will use distributive law as follows:
-74ab2c-625a2c2-50a2b2c2=-74ab2c-50a2b2c2-625a2c2-50a2b2c2=-74×-50a×a2×b2×b2×c×c2-625-50a2×a2×b2×c2×c2=-74×-50a1+2b2+2c1+2-625-50a2+2b2c2+2=1752a3b4c3--12a4b2
=1752a3b4c3+12a4b2c4
Thus, the answer is 1752a3b4c3+12a4b2c4.
Q u e s t i o n : 5 1
Find the following product:
-827xyz32xyz2-94xy2z3
S o l u t i o n :
To find the product, we will use the distributive law in the following way:
-827xyz32xyz2-94xy2z3=-827xyz32xyz2--827xyz94xy2z3=-827×32x×x×y×y×z×z2--827×94x×x×y×y2×z×z3=-827×32x1+1y1+1z1+2--827×94x1+1y1+2z1+3=-84279×32x1+1y1+1z1+2--82
=-49x2y2z3+23x2y3z4
Thus, the answer is -49x2y2z3+23x2y3z4.
Q u e s t i o n : 5 2
Find the following product:
-427xyz92x2yz-34xyz2
S o l u t i o n :
To find the product, we will use distributive law as follows:
-427xyz92x2yz-34xyz2=-427xyz92x2yz--427xyz34xyz2=-427×92x1+2y1+1z1+1--427×34x1+1y1+1z1+2=-42273×92x1+2y1+1z1+1--41279×34x1+1y1+1z1+2=-23x3y2z2+19x2y2z3
Thus, the answer is -23x3y2z2+19x2y2z3.
Q u e s t i o n : 5 3
Find the following product:
1.5x(10x
2
y - 100xy
2
)
S o l u t i o n :
To find the product, we will use distributive law as follows:
1.5x10x2y-100xy2=1.5x×10x2y-1.5x×100xy2=15x1+2y-150x1+1y2=15x3y-150x2y2
Thus, the answer is 15x3y-150x2y2.
Q u e s t i o n : 5 4
Find the following product:
4.1xy(1.1x - y)
S o l u t i o n :
To find the product, we will use distributive law as follows:
4.1xy1.1x-y=4.1xy×1.1x-4.1xy×y=4.1×1.1×xy×x-4.1xy×y=4.51x1+1y-4.1xy1+1=4.51x2y-4.1xy2
Thus, the answer is 4.51x2y-4.1xy2.
Q u e s t i o n : 5 5
Find the following product:
250.5xyxz+y10
S o l u t i o n :
To find the product, we will use distributive law as follows:
250.5xyxz+y10=250.5xy×xz+250.5xy×y10=250.5x1+1yz+25.05xy1+1=250.5x2yz+25.05xy2
Thus, the answer is 250.5x2yz+25.05xy2.
Q u e s t i o n : 5 6
Find the following product:
75x2y35xy2+25x
S o l u t i o n :
To find the product, we will use distributive law as follows:
75x2y35xy2+25x=75x2y×35xy2+75x2y×25x=2125x2+1y1+2+1425x2+1y=2125x3y3+1425x3y
Thus, the answer is 2125x3y3+1425x3y.
Q u e s t i o n : 5 7
Find the following product:
43a(a2 + b2 - 3c2)
S o l u t i o n :
To find the product, we will use distributive law as follows:
43aa2+b2-3c2=43a×a2+43a×b2-43a×3c2=43a1+2+43ab2-4ac2=43a3+43ab2-4ac2
Thus, the answer is 43a3+43ab2-4ac2.
Q u e s t i o n : 5 8
Find the product 24x
2
 (1 - 2x) and evaluate its value for x = 3.
S o l u t i o n :
To find the product, we will use distributive law as follows:
24x21-2x=24x2×1-24x2×2x=24x2-48x1+2=24x2-48x3
Substituting  x = 3 in the result, we get:
24x2-48x3=2432-4833=24×9-48×27=216-1296=-1080
Thus, the product is (24x2-48x3) and its value for x = 3 is (-1080).
Q u e s t i o n : 5 9
Find the product -3y(xy + y
2
) and find its value for x = 4 and y = 5.
S o l u t i o n :
To find the product, we will use distributive law as follows: ? -3yxy+y2=-3y×xy+-3y×y2=-3xy1+1-3y1+2=-3xy2-3y3
Substituting x = 4 and y = 5 in the result, we get:
-3xy2-3y3=-3452-353=-3425-3125=-300-375=-675
Thus, the product is (-3xy2-3y3), and its value for ?x = 4 and y = 5 is (-675).
Q u e s t i o n : 6 0
Multiply -32x2y3 by (2x - y) and verify the answer for x = 1 and y = 2.
S o l u t i o n :
To find the product, we will use distributive law as follows:
-32x2y3×2x-y=-32x2y3×2x--32x2y3×y=-3x2+1y3--32x2y3+1=-3x3y3+32x2y4
Substituting x = 1 and y = 2 in the result, we get:
-3x3y3+32x2y4=-31323+321224=-3×1×8+32×1×16=-24+24=0
Thus, the product is -3x3y3+32x2y4, and its value for ?x = 1 and y = 2 is 0.
Q u e s t i o n : 6 1
Multiply the monomial by the binomial and find the value of each for x = -1, y = 0.25 and z = 0.05:
i 15y
2
(2 - 3x)
ii -3x(y
2
 + z
2
)
iii z
2
(x - y)
iv xz(x
2
 + y
2
)
S o l u t i o n :
i To find the product, we will use distributive law as follows:
15y22-3x=15y2×2-15y2×3x=30y2-45xy2
Substituting x = -1 and y = 0.25 in the result, we get:
30y2-45xy2=300.252-45-10.252=30×0.0625-45×-1×0.0625=30×0.0625-45×-1×0.0625=1.875--2.8125=1.875+2.8125=4.6875
ii To find the product, we will use distributive law as follows:
-3xy2+z2=-3x×y2+-3x×z2=-3xy2-3xz2
Substituting x = -1, y = 0.25 ? and z = 0.05 ? in the result, we get:
-3xy2-3xz2=-3-10.252-3-10.052=-3-10.0625-3-10.0025=01875+0.0075=0.195
iii To find the product, we will use distributive law as follows:
z2x-y=z2×x-z2×y=xz2-yz2
Substituting x = -1, y = 0.25 ? and z = 0.05 ? in the result, we get: ? xz2-yz2=-10.052-0.250.052=-10.0025-0.250.0025=-0.0025-0.000625=-0.003125
iv To find the product, we will use distributive law as follows:
xzx2+y2=xz×x2+xz×y2=x3z+xy2z
Substituting x = -1, y = 0.25 ? and z = 0.05 ? in the result, we get: ? x3z+xy2z=-130.05+-10.2520.05=-10.05+-10.06250.05=-0.05-0.003125=-0.053125
Q u e s t i o n : 6 2
Simplify:
i 2x
2
(x
3
 - x) - 3x(x
4
 + 2x) - 2(x
4
 - 3x
2
)
ii x
3
y(x
2
 - 2x) + 2xy(x
3
 - x
4
)
iii 3a
2
 + 2(a + 2) - 3a(2a + 1)
iv x(x + 4) + 3x(2x
2
 - 1) + 4x
2
 + 4
v a(b - c) - b(c - a) - c(a - b)
vi a(b - c) + b(c - a) + c(a - b)
vii 4ab(a - b) - 6a
2
(b - b
2
) - 3b
2
(2a
2
 - a) + 2ab(b - a)
viii x
2
(x
2
 + 1) - x
3
(x + 1) - x(x
3
 - x)
ix 2a
2
 + 3a(1 - 2a
3
) + a(a + 1)
x a
2
(2a - 1) + 3a + a
3
 - 8
xi 32x2(x2-1)+14x2(x2+x)-34x(x3-1)
xii a
2
b(a - b
2)
 + ab
2
(4ab - 2a
2
) - a
3
b(1 - 2b)
xiii a
2
b(a
3
 - a + 1) - ab(a
4
 - 2a
2
 + 2a) - b (a
3
 - a
2
 - 1)
S o l u t i o n :
i To simplify, we will use distributive law as follows:
2x2x3-x-3xx4+2x-2x4-3x2=2x5-2x3-3x5-6x2-2x4+6x2=2x5-3x5-2x4-2x3-6x2+6x2=-x5-2x4-2x3
ii To simplify, we will use distributive law as follows: ? x3yx2-2x+2xyx3-x4=x5y-2x4y+2x4y-2x5y=x5y-2x5y-2x4y+2x4y=-x5y
iii To simplify, we will use distributive law as follows: ? 3a2+2a+2-3a2a+1=3a2+2a+4-6a2-3a=3a2-6a2+2a-3a+4=-3a2-a+4
iv To simplify, we will use distributive law as follows:
xx+4+3x2x2-1+4x2+4=x2+4x+6x3-3x+4x2+4=x2+4x2+4x-3x+6x3+4=5x2+x+6x3+4
v To simplify, we will use distributive law as follows: ? ab-c-bc-a-ca-b=ab-ac-bc+ba-ca+cb=ab+ba-ac-ca-bc+cb=2ab-2ac
vi To simplify, we will use distributive law as follows: ? ab-c+bc-a+ca-b=ab-ac+bc-ba+ca-cb   =ab-ba-ac+ca+bc-cb=0
vii To simplify, we will use distributive law as follows: ? 4aba-b-6a2b-b2-3b22a2-a+2abb-a=4a2b-4ab2-6a2b+6a2b2-6b2a2+3b2a+2ab2-2a2b=4a2b-6a2b-2a2b-4ab2+3b2a+2ab2+6a2b2-6b2a2=-4a2b+ab2
viii To simplify, we will use distributive law as follows: ? x2x2+1-x3x+1-xx3-x=x4+x2-x4-x3-x4+x2=x4-x4-x4-x3+x2+x2=-x4-x3+2x2
ix To simplify, we will use distributive law as follows: ? 2a2+3a1-2a3+aa+1=2a2+3a-6a4+a2+a=2a2+a2+3a+a-6a4=3a2+4a-6a4
x To simplify, we will use distributive law as follows: ? a22a-1+3a+a3-8=2a3-a2+3a+a3-8=2a3+a3-a2+3a-8=3a3-a2+3a-8
xi To simplify, we will use distributive law as follows: ? 32x2x2-1+14x2x2+x-34xx3-1=32x4-32x2+14x4+14x3-34x4+34x=32x4+14x4-34x4+14x3-32x2+34x=6+1-34x4+14x3-32x2+34x=x4+14x3-32x2+34x
xii To simplify, we will use distributive law as follows:
a2ba-b2+ab24ab-2a2-a3b1-2b=a3b-a2b3+4a2b3-2a3b2-a3b+2a3b2=a3b-a3b-a2b3+4a2b3-2a3b2+2a3b2=3a2b3
xiii To simplify, we will use distributive law as follows: ? a2ba3-a+1-aba4-2a2+2a-ba3-a2-1=a5b-a3b+a2b-a5b+2a3b-2a2b-a3b+a2b+b=a5b-a5b-a3b+2a3b-a3b+a2b-2a2b+a2b+b=b