Given:
[tex]x^y=z[/tex]Required:
A) We need to find the value of z when x =3 and y=4.
B) We need to find the value of x when y=3 and z =125.
C) We need to find whether the value of x is rational when y =2 and z=2.
D) We need to find the value of y and z when x=8.
Explanation:
A)
Substitute x =3 and y =4 in the given equation.
[tex]3^4=z[/tex][tex]3\times3\times3\times3=z[/tex][tex]z=81[/tex]Answer:
[tex]z=81[/tex]B)
Substitute y =3 and z=125 in the given equation.
[tex]x^3=125[/tex][tex]\text{Use }125=5^3.[/tex][tex]x^3=5^3[/tex][tex]\text{We know that if }a^n=b^n\text{ then }a=b.[/tex][tex]x=5[/tex]Answer:
[tex]x=5[/tex]C)
The rewrite of the given equation is
[tex]\sqrt[y]{z}=x[/tex]Substitute y =2 and z=2 in the equation.
[tex]\sqrt[2]{2}=x[/tex]The integer 2 is not a perfect square.
The square root of 2 is an irrational number.
The value of x is not a rational number.
Answer:
The value of x is not a rational number.
D)
[tex]\sqrt[y]{z}=x[/tex]Substitute x=8 in the equation.
[tex]\sqrt[y]{z}=8[/tex]Let y=2 and substitute in the equation.
[tex]\sqrt[2]{z}=8[/tex]Raise the power of both sides of the equation to 2.
[tex]z=8^2[/tex][tex]z=64[/tex]The value of y =2 and the value of z=64 make the equation true when x =8.
Answer:
The value of y =2 and the value of z=64 make the equation true when x =8.
