Given that (24z^3)^2/27x12z = 2^a 3^b z^c ,evaluate a,b and c.
is that
(24z^3)^2/27 * 12z
or
(24z^3)^2/(27 * 12z)
If the latter, then that is
(24^2 z^6)/(27 * 12 z)
= (8^2*3^2)/(3^3*2^2*3) * z^6/z
= (2^6 3^2)/(2^2 3^4) z^5
now finish it off
Well, let's break it down and have some fun with math!
First, let's simplify the expression: (24z^3)^2/27x12z
Now, (24z^3)^2 = 24^2 * z^6 = 576z^6
Dividing by 27 gives us: 576z^6 / 27x12z
Now, simplifying further, we get: (576 / 27) * (z^6 / z) * (1 / 12x)
Solving the numbers gives us: (24) * (z^5) * (1 / 12x)
Simplifying even more, we have: 2z^5 / x
So, now we need to express this in the form of 2^a * 3^b * z^c
In our expression, there's no 3, but we have 2z^5, which means a = 1, b = 0 (since there's no 3), and c = 5.
So, finally, a = 1, b = 0, and c = 5. Ta-da!
To evaluate the values of a, b, and c, we need to simplify the expression first.
Let's begin by simplifying (24z^3)^2:
(24z^3)^2 = (24^2)(z^3)^2 = 576z^6.
Now, let's simplify the whole expression:
(576z^6)/(27x12z) = 576z^6 / (27*12z) = 576z^6 / 324z = (576/324)(z^6/z) = (2/3)(z^5).
Now, compare the simplified expression with 2^a 3^b z^c:
2^a 3^b z^c = (2/3)(z^5).
From the expression, we can see that:
a = 1 (since 2^1 = 2)
b = 0 (since 3^0 = 1)
c = 5.
Therefore, a = 1, b = 0, and c = 5.
To evaluate the values of a, b, and c, we need to simplify the given expression first.
We can simplify the expression step by step:
1. Start with (24z^3)^2/27x12z.
2. Square the term inside the parentheses: 24^2 * (z^3)^2 / 27x12z.
3. Simplify the exponents: 576z^6 / 27x12z.
4. Simplify the numbers: (576 / 27) * (z^6 / x12z).
5. Simplify 576 / 27: 21.333...
6. Simplify z^6 / x12z: z^(6-1) / x12 = z^5 / x12.
Therefore, the simplified expression is 21.333... * z^5 / x12.
Now that we have the expression in the form 2^a 3^b z^c, we can determine the values of a, b, and c as follows:
a = 21 (since 21.333... is closest to 21)
b = 0 (since there are no 3's in the expression)
c = 5 (since the exponent of z is 5)
Hence, a = 21, b = 0, and c = 5.