Respuesta :
Answer:
Therefore the value of a is 3.78.
Step-by-step explanation:
Indices Rule:
- [tex](a^m)^n=a^{mn}[/tex]
- [tex]a^m.a^n=a^{m+n}[/tex]
- [tex]\frac{a^m}{a^n}=a^{m-n}[/tex]
- [tex]\sqrt[n]{a} =a^{\frac 1n}[/tex]
- [tex](\sqrt[n]{a} )^m=a^\frac mn[/tex]
Given that,
(a,b) satisfies the two equations
[tex]ab^4=384[/tex] .........(1)
and
[tex]a^2b^5=4608[/tex].........(2)
From equation (1) we get
[tex]ab^4=384[/tex]
Squaring both sides
[tex]\Rightarrow (ab^4)^2=(384)^2[/tex]
[tex]\Rightarrow a^2b^8=147,456[/tex]......(3)
Divide equation (3) by (2) we get
[tex]\frac{a^2b^8}{a^2b^5}=\frac{147,456}{4608}[/tex]
[tex]\Rightarrow a^{2-2}b^{8-5}=32[/tex]
[tex]\Rightarrow b^3=32[/tex]
[tex]\Rightarrow b=\sqrt[3] {32}[/tex]
Now plug the value of b in equation (1)
[tex]ab^4=384[/tex]
[tex]\Rightarrow a(\sqrt[3]{32})^4=384[/tex]
[tex]\Rightarrow a(32)^\frac43=384[/tex]
[tex]\Rightarrow a=\frac{384}{(32)^\frac43}[/tex]
[tex]\Rightarrow a\approx 3.78[/tex]
Therefore the value of a is 3.78.
