Help with Math Problems
Math Only Math is there to help with math problems from basic to
intricate problems. Care has been taken to help with math problems in
such a way that students can understand each and every step.
1. Simplify: 5n(2n^{3}+n^{2}+8)+n(4-n).
Solution:
5n(2n
^{3}+n
^{2}+8)+n(4-n).
= 5n × 2n
^{3} + 5n × n
^{2} + 5n × 8 + n × 4 - n × n.
= 10n
^{4} + 5n
^{3} + 40n + 4n – n
^{2}.
= 10n
^{4} + 5n
^{3} + 44n – n
^{2}.
= 10n
^{4} + 5n
^{3} – n
^{2} + 44n.
Answer: 10n^{4} + 5n^{3} – n^{2} + 44n
2. Simplify: 4y(y^{2}-8y+6)-3(2y^{3}-5y^{2}+2)
Solution:
4y(y
^{2}-8y+6)-3(2y
^{3}-5y
^{2}+2).
= 4y
^{3} - 32y
^{2} + 24y – 6y
^{3} + 15y
^{2} – 6.
= 4y
^{3} – 6y
^{3} - 32y
^{2} + 15y
^{2} + 24y – 6.
= -2y
^{3} – 17y
^{2} + 24y – 6.
Answer: -2y^{3} – 17y^{2} + 24y – 6
3. Find the area, diameter = 12m.
Solution:
Diameter = 12 meter.
Radius = diameter/2.
= 12/2.
= 6 meter.
Area of a circle = πr
^{2}.
Here, pi (π) = 3.14 meter, radius (r) = 6.
Area of a circle = 3.14 × 6 × 6..
= 3.14 × 36.
= 113.04 square meter or 113.04 m
^{2}.
Answer: 113.04 square meter or 113.04 m^{2}
4. Find the area, radius = 8cm.
Solution:
Given, Radius = 8 cm.
Area of a circle = πr
^{2}.
Area of a circle = 3.14 × 8 × 8 [We know, pi (π) = 3.14].
= 3.14 × 64.
= 200.96 square cm or 200.96 cm
^{2}.
Answer: 200.96 square cm or 200.96 cm^{2}.
5. Simplify: 4x^{2}(x+2)+3x(5x^{2}+2x-6)-5(3x^{2}-4x).
Solution:
4x
^{2}(x+2) +3x(5x
^{2}+2x-6) -5(3x
^{2}-4x).
= 4x
^{3} + 8x
^{2} + 15x
^{3} + 6x
^{2} – 18x – 15x
^{2} + 20x.
= 4x
^{3} + 15x
^{3} + 8x
^{2} + 6x
^{2} – 15x
^{2} – 18x + 20x.
= 19x
^{3} - x
^{2} + 2x.
Answer: 19x^{3} - x^{2} + 2x
6. Factoring with repeated of the difference of square formula. Factor the answer completely.
3y^{4} – 3u^{4}y^{4}.
Solution:
3y
^{4} – 3u
^{4}y
^{4}.
= 3y
^{4}(1 – u
^{4}).
= 3y
^{4}(1 – (u
^{2})
^{2}).
= 3y
^{4}(1 + u
^{2})( 1 - u
^{2}) [since we know a
^{2} – b
^{2} = (a + b)(a – b)].
= 3y
^{4}(1 + u
^{2})( 1 + u) ( 1 - u).
Answer: 3y^{4}(1 + u^{2})( 1 + u) ( 1 - u).
7. Factoring with repeated of the difference of square formula. Factor the answer completely.
u^{4}x^{3} – 81x^{3}.
Solution:
u
^{4}x
^{3} – 81x
^{3}.
= x
^{3}(u
^{4} – 81).
= x
^{3}((u
^{2})
^{2} – 9
^{2}).
= x
^{3}(u
^{2} + 9)(u
^{2} – 9).
= x
^{3}(u
^{2} + 9)(u
^{2} – 3
^{2}).
= x
^{3}(u
^{2} + 9)(u + 3)(u – 3).
Answer: x^{3}(u^{2} + 9)(u + 3)(u – 3).
8. Factoring with repeated of the difference of square formula. Factor the answer completely.
2w^{2} – 2v^{4}w^{2}.
Solution:
2w
^{2} – 2v
^{4}w
^{2}.
= 2w
^{2} (1 – v
^{4}).
= 2w
^{2} (1 – (v
^{2})
^{2}).
= 2w
^{2} (1 + v
^{2})(1 - v
^{2}).
= 2w
^{2} (1 + v
^{2})(1 + v)(1 – v).
Answer: 2w^{2} (1 + v^{2})(1 + v)(1 – v).
9. Factoring with repeated of the difference of square formula. Factor the answer completely.
3u^{2}x^{4} – 48u^{2}.
Solution:
3u
^{2}x
^{4} – 48u
^{2}.
= 3u
^{2}(x
^{4} – 16).
= 3u
^{2}((x
^{2})
^{2} – 4
^{2}).
= 3u
^{2}(x
^{2} +4)(x
^{2} – 4).
= 3u
^{2}(x
^{2} +4)(x
^{2} – 2
^{2}).
=3u
^{2}(x
^{2} +4)(x + 2)(x – 2).
Answer: 3u^{2}(x^{2} +4)(x + 2)(x – 2).
10. Factoring a sum or difference of two cubes.
Factor: 64 + u^{3}.
Solution:
64 + u
^{3}.
= 4
^{3} + u
^{3}.
= (4 + u)(4
^{2} – 4.u + u
^{2}).
= (4 + u)(16 – 4u + u
^{2}).
Answer: (4 + u)(16 – 4u + u^{2})
11. Factoring a sum or difference of two cubes.
Factor: 125 - 8u^{3}
Solution:
125 - 8u
^{3}.
= 5
^{3} – (2u)
^{3}.
= (5 – 2u)(5
^{2} + 5.2.u + (2u)
^{2} ).
= (5 – 2u)(25 + 10u + 4u
^{2}).
Answer: (5 – 2u)(25 + 10u + 4u^{2})
12. Solving a linear equation with several occurrences of the variable, solve for w. Simplify your answer as much as possible.
(7w + 6)/6 + (9w +8)/2 = 22
Solution:
(7w + 6)/6 + (9w +8)/2 = 22
or, [7w + 6 + 3(9w + 8)]/6 = 22
or, 7w + 6 + 27w + 24 = 132
or, 34w + 30 = 132
or, 34w = 132 - 30
or, 34w = 102
or, w = 102/34
Therefore, w = 3
Answer: w = 3
13. Find the value of p: 2p+8p-6=7p+2p
Solution:
2p + 8p - 6 = 7p + 2p
or, 10p – 6 = 9p
or, 10p – 6 – 9p = 9p – 9p
[Subtract 9p from both the sides]
or, p – 6 = 0
or, p - 6 + 6 = 6
[Add 6 to both the sides]
Therefore, p = 6.
Answer: p = 6
14. Simplify: (24sp)/(3s)
Solution:
(24sp)/(3s).
= 8p.
Answer: 8p
15. Rewrite 3/8 as a decimal
Solution:
3/8 is a fraction; we need to change 3/8 to decimal number by dividing.
Answer: 0.375
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