Question

If $x=y^a,y=z^b,z=x^c$ then find abc. If a = 12, b = $\frac{1}{288}$, then find the value of c. [4 MARKS]

Answer 1

Steps: 1 Marks
Application: 1 Mark
Value of abc: 1 Mark
Value of c: 1 Mark

It is given that
$x=y^a,y=z^b,z=x^c$

$x=y^a$

$=(z^b{)}^a$ $(\because y=z^b)$

$=z^{ab}$

$=(x^c{)}^{ab}$ $(\because x^c=z)$

$x=x^{abc}$ $\because \left(\right(a^m{)}^n=a^{mn})$

$\Rightarrow x^1=x^{abc}$

As bases are equal, exponents are equal.

$\Rightarrow abc=1$

Now, it is also given that,
a = 12, b = $\frac{1}{288}$
a$\times$b$\times$c = 1
c = $\frac{1}{a\times b}$

On substituting the values we get:
c = $\frac{1}{12\times \frac{1}{288}}$

c = $\frac{288}{12}$

c = $24$
Hence, the value of c is $24$.