1)
(5^1/5)^5 = 25^(1/2) = 5
x =1/2
2) (8^1/3)^2 = 4^1 = 4
x = 1
Answer:
1) [tex]x=\frac{1}{2}[/tex]
2) [tex]x=1[/tex]
Step-by-step explanation:
We are asked to find value of x for each of our given expressions.
1). [tex](5^{\frac{1}{5}})^5=25^x[/tex]
Substitute [tex]25=5^2[/tex]:
[tex](5^{\frac{1}{5}})^5=(5^2)^x[/tex]
Using exponent property [tex](a^m)^n=a^{m\cdot n}[/tex], we will get:
[tex]5^{\frac{1}{5}\cdot 5}=5^{2\cdot x}[/tex]
[tex]5^{1}=5^{2x}[/tex]
We know when [tex]a^n=a^{m}[/tex], then [tex]n=m[/tex]. Since base of both exponents is equal, so we can equate them as:
[tex]1=2x[/tex]
[tex]2x=1[/tex]
[tex]\frac{2x}{2}=\frac{1}{2}[/tex]
[tex]x=\frac{1}{2}[/tex]
Therefore, the value of x is [tex]\frac{1}{2}[/tex].
(2). [tex](8^{\frac{1}{3}})^2=4^x[/tex]
Using exponent property [tex](a^m)^n=a^{m\cdot n}[/tex], we will get:
[tex]8^{\frac{1}{3}\times 2}=4^x[/tex]
[tex]8^{\frac{2}{3}}=4^x[/tex]
Substitute [tex]8=2^2[/tex] and 4 as Substitute [tex]4=2^2[/tex]:
[tex](2^3)^{\frac{2}{3}}=(2^2)^x[/tex]
Using exponent property [tex](a^m)^n=a^{m\cdot n}[/tex], we will get:
[tex]2^{3\times\frac{2}{3}}=2^{2*x}[/tex]
[tex]2^{2}=2^{2x}[/tex]
We know when [tex]a^n=a^{m}[/tex], then [tex]n=m[/tex]. Since base of both exponents is equal, so we can equate them as:
[tex]2=2x[/tex]
[tex]2x=2[/tex]
[tex]\frac{2x}{2}=\frac{2}{2}[/tex]
[tex]x=1[/tex]
Therefore, the value of x is 1.
