Fully simplify
z^6\cdotz^7)^5/(z^5\cdotz^2)^3
First, let's simplify the numerator:
(z^6 * z^7)^5
= z^13^5
Next, let's simplify the denominator:
(z^5 * z^2)^3
= z^7^3
Now, let's simplify the expression as a whole:
(z^13^5) / (z^7^3)
= z^65 / z^21
To simplify further, we can subtract the exponents:
= z^(65-21)
= z^44
Therefore, the fully simplified expression is z^44.
To fully simplify the expression (z^6 * z^7)^5 / (z^5 * z^2)^3, we can start by applying the power of a product rule and simplifying each term separately.
First, let's simplify the numerator (z^6 * z^7)^5:
(z^6 * z^7)^5 = z^(6*5) * z^(7*5) = z^30 * z^35 = z^(30+35) = z^65
Now let's simplify the denominator (z^5 * z^2)^3:
(z^5 * z^2)^3 = z^(5*3) * z^(2*3) = z^15 * z^6 = z^(15+6) = z^21
Therefore, the fully simplified expression is z^65 / z^21.
To fully simplify the expression (z^6 * z^7)^5 / (z^5 * z^2)^3, we can use the properties of exponents and simplify the individual terms before combining them.
First, let's simplify the expression inside the parentheses for both the numerator and denominator.
In the numerator, we have z^6 * z^7. When we multiply two numbers with the same base, we add their exponents. Therefore, z^6 * z^7 can be rewritten as z^(6 + 7) = z^13.
In the denominator, we have z^5 * z^2. Similarly, z^5 * z^2 can be rewritten as z^(5 + 2) = z^7.
Now that we have simplified the terms inside the parentheses, we can rewrite the expression as (z^13)^5 / (z^7)^3.
Next, let's simplify the exponents. When we raise a power to another power, we multiply the exponents. Hence, (z^13)^5 can be expressed as z^(13 * 5) = z^65.
Similarly, (z^7)^3 can be written as z^(7 * 3) = z^21.
Now, the expression becomes z^65 / z^21.
Finally, to simplify this division, we subtract the exponents. So, z^65 / z^21 can be expressed as z^(65 - 21) = z^44.
Therefore, the fully simplified expression is z^44.