Answer:
Step-by-step explanation:
1). [tex]2^{x^{2} +2x}=\frac{1}{2}[/tex]
[tex]2^{x^{2} +2x}=2^{-1}[/tex]
Comparing exponents on both the sides.
x² + 2x = -1
x² + 2x + 1 = 0
(x + 1)² = 0
x = -1
2). [tex]2^{x-3}=8^{x+1}[/tex]
[tex]2^{x-3}=(2^{3})^{x+1}[/tex]
[tex]2^{x-3}=(2)^{3(x+1)}[/tex]
By comparing exponents,
x - 3 = 3(x + 1)
x - 3 = 3x + 3
3x - x = 3 + 3
2x = 6
x = 3
3). [tex]\sqrt{a^{5-k}}=a^{3-k}[/tex]
[tex]a^{\frac{5-k}{2}}=a^{3-k}[/tex]
By comparing exponents,
[tex]\frac{5-k}{2}=3-k[/tex]
5 - k = 2(3 - k)
5 - k = 6 - 2k
2k - k = 6 - 5
k = 1
4). [tex]3^{x+3}=27^{(2x+6)}[/tex]
[tex]3^{x+3}=(3^3)^{2x+6}[/tex]
[tex]3^{x+3}=(3)^{3(2x+6)}[/tex]
By comparing exponents,
x + 3 = 3(2x + 6)
x + 3 = 6x + 18
6x - x = 3 - 18
5x = - 15
x = -3
5). [tex]5^{x-2}=1[/tex]
[tex]5^{(x-2)}=5^{0}[/tex]
x - 2 = 0
x = 2
