Students and Teacher's Forum

Students and Teachers Forum

Here, 4.3x+1 – 9x+1 = 3 Or, 4.3x. 31 – 9x. 91 = 3 Or, 12.3x – 9.(32)x = 3 Or, 12.3x – 9.(3x)2 = 3. Let , 3x = a Then, 12.3x – 9.(3x)2 = 3 Or, .....

2. Prove that:

[(p+ 1/q)^m (p- 1/q)ⁿ] / [(q+ 1/p)^m (q- 1/p)ⁿ ]= [p / q]^(m+n)

LHS = [(p+ 1/q)m (p- 1/q)n] / [(q+ 1/p)m (q- 1/p)n ] = (((pq+1)/q)m ( (pq-1)/q)n)/(( (pq+1)/p)m ((pq-1)/p)n ) = (((pq+1)/q)/((pq+1)/p))m ⨉ .....

3. Find the value of:

(5ⁿ⁺³- 55×5^n-1) / (5ⁿ⁺²+ 89×5ⁿ )

Here, (5n+3 - 55×5n-1) / (5n+2 + 89×5n ) = (5n×53-55×5n/5) / (5n×52+89×5n ) = (5n×53-11×5n) / (5n×52+89×5n ) = {5n (53-11)} / {5n (52+89)} = .....

4. Prove that:
((x^(a+b))³(x^(b+c))³ (x^(c+a))³)/(x^a x^b x^c)⁶ = 1

Here, LHS = ((x(a+b))3 (x(b+c))3 (x(c+a))3)/(x(a+b+c))6 = ((x(3a+3b))(x(3b+3c))(x(3c+3a)) )/(x(a+b+c))6 = x(3a+3b+3b+3c+3c+3a)/(x(a+b+c))6 .....

5. Prove that:
a = b^x, b = c^y and c = a^z, prove that xyz = 1.

Here, Given, a = bx, b = cy and c = az To prove: xyz = 1. Here, a = bx Or, a = (cy)x Or, a = cyx Or, a = (az)yx Or, a = axyz Equating, 1 = xyz So, xyz = .....

6. Find the value of:

1/(1+x^(l-m)) +1/(1+x^(m-l) )

Here, 1/(1+x(l-m) ) +1/(1+x(m-l) ) = 1/(1+xl / xm ) +1/(1+xm / xl ) = 1/((xm+xl)/xm ) +1/(xl+xm/xl ) = xm/(xm+xl ) +xl/(xm+xl ) = (xm+xl)/(xm+xl ) = 1 .....

7. Solve for x:

2^x- 4^x-1 = 1

Here, 2x - 4x-1 = 1 Or, 2x - 4x. 4-1 = 1 Or, 2x - 4x/4 = 1 Or, 2x – (22)x/4 = 1 Or, 2x – (2x)2/4 = 1. Let, 2x = .....

8. Show that:

(x^(1/(p-q))^(1/(p-r) ⨉ (x^(1/(q-r))^(1/(q-p) ⨉(x^(1/(r-p))^(1/(r-q) = 1

LHS = (x(1/(p-q ))(1/(p-r) ⨉ (x(1/(q-r))(1/(q-p) ⨉ (x(1/(r-p))(1/(r-q) = x(1/(p-q)(p-r) ) ⨉ x(1/(q-r)(q-p) ⨉ x(1/(r-p)(r-q)) = x(1/(p-q)(p-r)+ .....

9. Solve for x:

5.4^x+1 = 64 + 16^x

Here, 5.4x+1 = 64 + 16x Or, 5.4x. 41 = 64 + (42)x Or, 20.4x = 64 + (42)x Let, 4x = a Then, 20a = 64 + a2 Or, a2 – 20a + 64 = 0 Or, a2 – (16+4)a + 64 = 0 Or, a2 – 16a – 4a + 64 = .....

10. If a + b + c = 0, prove that :

1/(1+ x^a+ x^-b) + 1/(1+ x^b+ x^-c ) )+ 1/(1+ x^c+ x^-a) = 1

Given, a + b + c = 0 To prove: 1/(1+ xa + x-b ) + 1/(1+ xb + x-c ) )+ 1/(1+ xc + x-a ) = 1 LHS = 1/(1+ xa + x-b ) + 1/(1+ xb + x-c ) )+ 1/(1+ xc + x-a ) = .....

11. Find the value of:

(1/27)^-4/3× (27)^1/3

Here, (1/27)-4/3 × (27)1/3 = (1/33 )-4/3 × (33 )1/3 = 3-3×-4/3 × 33×1/3 = 34×3 = 81×3 = 243 .....

12. Solve for x:

4^x+ 128 = 6.2^x+2

Here, 4x+ 128 = 6.2x+2 Or, (22)x+ 128 = 6.2x. 22 Or, (2x)2+ 128 = 6.2x. 4 Or, (2x)2+ 128 = 24.2x Let, 2x = a, Then, a2 + 128 = 24a Or, a2 – 24a + 128 = 0 Or, a2 – (16+8)a + 128 = .....

13. Solve for x:

5^x + 1/5^{^x} = 26/5

Here, 5x + 1/5x = 26/5 Or, 5x.5x+ 1/5x = 26/5 Or, 52x+ 1/5x = 26/5 Or, 5x(2)+1/5x = 26/5 Let 5x = a, Then, a2+1.a = 265 Or, 5 (a2 + 1) = 26a Or, 5a2 + 5 = 26a Or, 5a2 – 26a + 5 = 0 Or, .....

14. Find the value of:

(13^2x+1+5×169^x) / (9×169^x )

Here, (132x+1 +5×169x) / (9×169x ) = (132x ×131 + 5×132x) / (9×132x ) = {132x (13+5)} / (9×132x ) = 18/9 = 2 .....

15. Find the value of:
(x+y)-83 × x+y23

Here, (x+y)-83 × x+y23 = x+y-83 × x+y23 = x+y-83+23 = x+y-8+23 = x+y-63 = x+y-2 = 1(x+y)2 .....

16. Find the value of:

y-2y43 × y4x2

Here, y-2y43 × y4x2 = y-2×y23× y4x = y(-2+2)×13 × y4×12x =y0×13× y2x = 1×y2x = x y2 . .....

17. Simplify:
64x3÷27a-3-23

Here, 64x3÷27a-3-23 = 64x327a-3-23 = 33a-343x323 = 3a-14x2 = 34ax2 .....

18. Simplify: (278)-1/3[(8116)14÷425-12]

Here, (278)-1/3[(8116)14÷425-12] =( 3323)-1/3[(3424)1/4×(2252)1/2 = 23×32×25 = 25 .....

19. If pqr = 1, prove that:

11+p+q-1+11+q+r-1+11+r+p-1=1

We have,pqr = 1 Now, L.H.S= 11+p+q-1+11+q+r-1+11+r+p-1 =11+p+1q+11+q+1r+11+r+1p =1q+pq+1q +1r+rq+1r +1p+pr+1p =q q+pq+1+rr+rq+1+ p p+pr+1 =q q+pq+1+pp×rr+rq+1+qq×p p+pr+1 =q q+pq+1+prpr+pqr+p+pq pq+pqr+q =q q+pq+1+prpr+1+p+pq .....

20. If p + q + r = 0, prove that:

11+xp+x-q+11+xq+x-r+11+xr+x-p = 1

Here, p + q + r = 0 We have, L.H.S = .....

21. Prove that the value of x obtained by solving 4×3x+1-9x=27 also satisfy the equation 32x-4×3x+1+813=0

Here, We have, 4×3x+1-9x=27 ………………….eqn (1) or, 4×3x.31-32x=27 or, 12.3x-3x2=27 or, 3x2-12.3x+27=0 or,3x2-9.3x-3.3x+27=0 or, 3x3x-9-33x-9=0 or,3x-33x-9=0 Either, 3x=3 ∴x=1 Or, .....

22. Prove that:

ax/yay/xx+y ×ay/zaz/yy+z ×az/xax/zz+x = 1

LHS = ax/yay/xx+y × ay/zaz/yy+z × az/xax/zz+x = axy-yxx+y × ayz-zyy+z ×azx-xzz+x = ax2-y2xyx+y × ay2-z2yzy+z × az2-x2xzz+x = a x2-y2xy(x+y) .....

23. Solve:
3x+2 + 13x-2 = 30

Here, 3x+2 + 13x-2 = 30 or, 3x-2+x+2+ 1 3x-2 = 30 Or, 32x + 1= 30 ⨉ 3x ⨉ (1/9) Let 3x = a, So, The equation becomes, a2 + 1= 30a/9 Or,a2 = 10a / 3 Or, 3a2 + 3 – 10a=0 Or, (3a .....

24. Simplify:
27x6y33÷81x8y44

Here, 27x6y33÷81x8y44 = 27x6y3381x8y44 = 33x6y3334x8y44 = 3x2y3x2y = .....

25. Simplify:
x-1x-1+y-1+y-1x-1+y-1

Here, x-1x-1+y-1+y-1x-1+y-1 = x-1+y-1x-1+y-1 = .....

26. Find the value of:

5^-n×25^2n-2/ {5^3n-2× (5×2)^-1 }

Here, 5-n × 52(2n-2) / (53n-2 × 5-1 × 2-1 ) = 5(-n+4n-4) / [5(3n-2-1) × 2-1] = 5(3n-4) /[ 5(3n-3) × 2-1] = 2/5(3n-3-3n+4) = 2/5 .....

27. Prove that:
{(a^x . a^y)^x-y ÷ (a^z/a^y )^y+z} ⨉ (a^z/a^x )^z+x = 1

Here, LHS = { (ax . ay)x-y ÷(azay)y+z }×(azax)z+x = { (ax+y)x-y ÷(az-y)y+z } ×(az-x)z+x = a (x+y)(x-y) ÷a(z-y)(z+y) ×a(z-x)(z+x) .....

28. If x = a^p+q b^r, y = a^q+rb^p and z = a^r+pb^q, prove that x^p-q.y^q-r.z^r-p = 1.

Given, x = ap+q br y = aq+rbp z = ar+pbq To prove: xp-q.yq-r.zr-p = 1. LHS = (ap+q br)p-q.( aq+rbp)q-r.(ar+pbq)r-p = .....

29. Find the value of:
a-b-1 (b-2-a-2)

Here, (a-b)-1 (b-2- a-2) = 1(a-b)1b2- 1a2 = 1(a-b)×a2-b2a2b2 = 1(a-b)× a+b(a-b)a2b2 = a+ba2b2 .....

30. Find the value of:
( 14⁶×15⁵) / (35⁶×6⁵ )

Here, (146×155) / (356×65 ) = {(2×7)6 × (3×5)5} / {(5×7)6 × (2×3)5 } = (26×76×35×55) / (56×76×25×35 ) = (26×55) / (56×25 ) = .....

31. Find the value of:

(x^(2m+3n) .x^(3m-6n)) / (x^(m+2n) . x^(4m-4n))

Here, (x(2m+3n) .x(3m-6n)) / (x(m+2n) . x(4m-4n) ) = x(2m+3n+3m-6n) ) / x(m+2n+4m-4n) = x(5m-3n) /x(5m-2n) = x(5m-3n-5m+2n) = x-n = 1/xn .....

32. If x = a^1/3 – a^-1/3, show that x³ + 3x = a - 1/a .

Given, x = a1/3 – a-1/3 To prove: x3 + 3x = a - 1/a LHS = x3 + 3x = (a1/3 – a-1/3)3 + 3 (a1/3 – a-1/3) = (a1/3)3 – 3.(a1/3)2 .( a-1/3) + 3.(a1/3) .( a-1/3)2 – ( a-1/3)3 + 3 .....

33. Simplify:
112x+1-6×121x5×121x

Here, 112x+1-6×121x5×121x = 112x.11-6×112x5×112x = 112x11-6112x.5 = .....

34. Simplify:
m+(mn2)13 +(m2n)13m-n × [1 -n13m13]

Here, = m13 m 23+ n23 + m13.n 13m3. 13-n3 .13 ×m13-n13m13 = m13 m 23+ n23 + m13.n 13(m13-n13) m 23+ n23 + m13.n 13 ×m13-n13m13 = .....

35. If a + b + c = 0, prove that:

11+xa+x-b+11+xb+x-c+11+xc+x-a

Here, a + b + c = 0 We have, L.H.S = .....

36. Solve :

2^x+3.3^x+4 = 18

Here, 2x+3.3x+4 = 18 Or,2x+3.3x+4 = 21 × 32 Or,2x+3-1 . 3x+4-2 = 1 Or,2x+2 . 3x+2 = 1 Or,(2.3)x+2 = 1 Or,6x+2 = 1 Or, 6x+2 = 60 Equating, x+2 = 0 Or, .....

37. Solve :

5 ⨉ 25^x+1 = (125)^2x-1

Here, 5 ⨉ 25x+1 = (125)2x-1 Or, 5 ⨉ (52)x+1 = (5)3(2x-1) Or, 5 ⨉ 52x+2 = 56x-3 Or, 51+2x+2 = 56x-3 Or, 53+2x = 56x-3 Equating, 3+2x = 6x-3 Or, 3+3 = 6x-2x Or, 6 = 4x Or, x = 6/4 Or, x = 3/2 ∴ x .....

38. Find the value of:

∛(3x⁷ y¹¹ z^-1 ) × ∛(72x^-1yz⁴ )

Here, ∛(3x7 y11 z-1 ) × ∛(72x-1yz4 ) = ∛((3x7 y11 z-1)×72x-1 yz4 ) = ∛(63 x7-1 y11+1 z-1+4 = (63 x6 y12 z3 )1/3 = (6x2 y4 z)3×1/3 = 6x2 y4 z . .....

39. Prove that:
[x^(l+m)/x^(l-n) ]^(m-n) ⨉ [x^(m+n)/x^(m-n) ]^(n-l) ⨉ [x^(n+l)/x^(n-m) ]^(l-m) = 1

Here, LHS = [x(l+m)/x(l-n) ](m-n) ⨉ [x(m+n)/x(m-n) ](n-l) ⨉ [x(n+l)/x(n-m) ](l-m) = (xl+m-l+n)m-n (xm+l-m+n)n-l (xn+l-n+m)l-m = (xm+n)m-n (xn+l)n-l (xl+m)l-m .....

40. Show that: (3^m+1+ 2 ⨉ 3^m-1 ) /( 3^m+3)- 6 ⨉ 3^m+1 ) = 11/27

Here, LHS = (3(m+1 )+ 2 ⨉ 3(m-1 ) )/(3(m+3 )- 6 ⨉ 3(m+1 ) ) = (3(m ). 3+ 2 ⨉ 3(m ). 3(-1 ) )/(3(m ). 3(3 )- 6 ⨉ 3(m ). 3(1 ) ) = .....

41. Prove that:
(x^(a+b)/x^{(b+ c)})^(a+c) ⨉ (x^(b+c)/x^(c+a) )^(b+a) ⨉ (x^(c+a)/x^(a+b) )^(c+b) = 1

Here, LHS = (x(a+b)/x(b+ c) )(a+c) ⨉ (x(b+c)/x(c+a) )(b+a) ⨉ (x(c+a)/x(a+b) )(c+b) = (xa + b – b – c) a+c (xb+c-c-a)b+a (xc+a-a-b)c+b = (xa-c)a+c .....

42. Find the value of:

∜[(48x¹¹ y⁹) / (3x³ y⁵ z^-4)]

Here, ∜[(48x11 y9) / (3x3 y5 z-4)] = [16x11-3y9-5z4 ]1/4 = [16x8 y4 z4]1/4 = [24 x8 y4 z4 ]1/4 = [2x2 yz]4×1/4 = 2x2yz . .....

43. Solve :

3^2x+3 – 2.9^x+1 = 1/3

Here, 32x+3 – 2.9x+1 = 1/3 Or, 32x+3 – 2.(32)x+1 = 1/3 Or, 32x+3 – 2.32x+2 = 1/3 Or, 32x. 33 – 2.32x.32 = 1/3 Or, 27. 32x – 2 . 9 .32x =1/3 Or, 27. 32x – 18 . 32x =1/3 Or, 32x (27 .....

44. Find the value of:

(b^x/c^x )^y. (c^y/a^y )^x. (a^x/b^x )^y

Here, (bx/cx )y. (cy/ay )x. (ax/bx )y = bxy/cxy . cxy/axy . axy/bxy = bxy-xy.axy-xy.cxy-xy = b0. a0.c0 = 1.1.1 = 1 .....

45. Find the value of:

(81)^-3/4 ×7

Here, (81)-3/4 ×7 = (34 )-3/4 ×7 = 34×-3/4 ×7 = 3-3 ×7 = 1/33 ×7 = 7/27 .....

46. Prove that:
(xpxq)p2+ pq+q2 ×(xqxr)q2+ qr+r2 (xrxp)r2+ rp+p2 = 1

Here, LHS = xpxqp2+ pq+q2 ×xqxrq2+ qr+r2 ×xrxpr2+ rp+p2 =xp-qp2+ pq+q2 ×xq-rq2+ qr+r2 ×xr-pr2+ rp+p2 = x(p-q)(p2+ pq+q2) ×xq-r(q2+ qr+r2) ×xr-p(r2+ .....

47. Solve for x:

9^x – 4.3^x + 3 = 0

Here, 9x – 4.3x + 3 = 0 Or, (32)x – 4.3x + 3 = 0 Or, (3x)2 – 4.3x + 3 = 0. Let, 3x = a Then, (3x)2 – 4.3x + 3 = 0 Or, a2 – 4a + 3 = 0 Or, a2 – (3+1)a + 3 = 0 Or, a2 .....

48. Solve :

9^2x+1 = 27^x+2

Here, 32(2x+1) = 33(x+2) Or,34x + 2 = 33x+6 Equating, 4x + 2 = 3x + 6 Or,4x – 3x = 6 – 2 Or,x = 4 ∴ x = 4 .....

49. Prove that:
11+ xa-b + 11+ xb-a = 1

LHS = 11+ xa-b + 11+ xb-a = 11+ xaxb + 11+ xbxa = 1xb+xaxb + 1xa+xbxa = xbxa+xb + xaxa+xb = xb+xaxa+xb = 1 .....

50. Simplify:
3n+2+5×3n3n×5-3n-1

Here, 3n+2+5×3n3n×5-3n-1 = 3n.9+5×3n3n×5-3n3 = 3n9+53n5-13 = .....