Math Master

x³ = -8
বা, x³ + 8 = 0
বা, x³ + 2³ = 0
(x + 2)(x² - 2x + 4) = 0
হয়,
x + 2 = 0
অথবা,
x² - 2x + 4 = 0

x - y = 3
(x - y)² = 9
বা, x² + y² - 2xy = 9
বা, 17 - 2xy = 9
2xy = 17 - 9 = 8
∴ xy = 4 = 4.1
∴ x = 4, y = 1 হলে
x - y = 3

ax - by = 0
বা, ax = by
∴ x = (b/a)y
আবার,
ay - bx = a² - b²
বা, ay - b.(b/a)y = a² - b²
বা, y(a - b²/a) = a² - b²
y × (a² - b²)/a = a² - b²
বা, y/a = 1
∴ y = a

x/y = 3
বা, x = 3y
আবার,
x + 2y = 15
বা, 3y + 2y = 15
বা, 5y = 15
∴ y = 3
∴ x = 3y
= 3.3
= 9

6/x = √2
বা, 6/√2 = x
x = 6/√2
x/√3 = 6/√2√3
= 6/√6
= (√6)²/√6
= √6

{(³√a)⁰}⁵
= {(a^1/3)⁰}⁵
= (a⁰)⁵
= 1⁵
= 1

c^r = a
বা, (b^q)^r = a
বা, b^qr = a
বা, (a^p)^qr = a
বা, a^pqr = a¹
∴ pqr = 1

x^-4 - 0.0001 = 0
বা, x^-4 = 0.0001
বা, 1/x⁴ = 1/10000
বা, x⁴ = 10000
বা, x⁴ = 10⁴
বা, x = 10
∴ x³ = 10³
= 1000

(³√5 × √2)⁶/³√512
= (5^1/3 × 2^1/2)⁶/³√8³
= (5² × 2³)/8
= (25 × 8)/8
= 25

10^{x + 2} = 10000
বা, 10^{x + 2} = 10⁴
বা, x + 2 = 4
∴ x = 2
∴ 2^{x + 3} = 2^{2 + 3}
= 2⁵
= 32

3^{2x - 1} = 3a^{2x - 2}
বা, 3^{2x - 1}/3a^{2x - 2} = 1
বা, 3^{2x - 2}/a^{2x - 2} = 1
বা, (3/a)^{2x - 2} = (3/a)⁰
বা, 2x - 2 = 0
বা, 2x = 2
∴ x = 1

x^y = y^x
বা, (2y)^y = y^2y
বা, 2^yy^y = y^2y
2^y = y^y
∴ y = 2
∴ x = 2y
= 2.2
= 4

log₂₇3√3 = log₂₇3^3/2
               = log₂₇27^1/2
               =1/2 log₂₇27
               = 1/2

ধরি,
ভিত্তি = a
∴ log_a400 = 4
বা, a⁴ = 400
বা, a⁴ = (2√5)⁴
∴ a = 2√5

log_ax = 1,
log_ay = 2
log_az = 3
∴ x = a¹ = a
∴ y = a²
∴ z = a³
log_a(x³y²/z) = log_a(a³.a⁴/a³)
= log_aa⁴
= 4log_aa
= 4.1
= 4

1/5log_x1024 = 1
বা, log_x1024 = 5
বা, x⁵ = 1024
= 4⁵
∴ x = 4