Fully simplify
(z^-7\cdotz^4)^2/z^3\cdotz^3)^3
To fully simplify the expression, we can start by simplifying the numerator and denominator separately, then rewrite the expression without negative exponents, and finally simplify any remaining exponents.
Starting with the numerator:
(z^-7 * z^4)^2 = z^-14 * z^8
Using the property of exponents that says z^a * z^b = z^(a+b), we can rewrite this as:
z^-14 * z^8 = z^(-14 + 8) = z^-6
moving to the denominator:
(z^3 * z^3)^3 = z^6 * z^6 = z^(6 + 6) = z^12
So, the fully simplified expression is:
(z^-7 * z^4)^2 / (z^3 * z^3)^3 = z^-6 / z^12 = z^(-6 - 12) = z^-18
To fully simplify the expression
((z^-7 · z^4)^2 / (z^3 · z^3)^3)
we can start by simplifying the numerator and then the denominator, and finally simplify the entire fraction.
Step 1: Simplify the numerator.
Inside the parentheses, we have an exponent rule for multiplying terms with the same base.
(z^-7 · z^4)^2 can be simplified as follows:
(z^-7 · z^4)^2 = z^(-7 · 2) · z^(4 · 2) = z^-14 · z^8
So the numerator becomes z^-14 · z^8.
Step 2: Simplify the denominator.
Similarly, for the denominator, we apply the same exponent rule to simplify (z^3 · z^3)^3.
(z^3 · z^3)^3 can be simplified as follows:
(z^3 · z^3)^3 = z^(3 · 3) · z^(3 · 3) = z^9 · z^9
So the denominator becomes z^9 · z^9.
Step 3: Simplify the entire expression.
Now, we can rewrite the expression with the simplified numerator and denominator:
((z^-7 · z^4)^2 / (z^3 · z^3)^3) = (z^-14 · z^8) / (z^9 · z^9)
To divide two terms with the same base, we subtract their exponents:
(z^-14 · z^8) / (z^9 · z^9) = z^(-14 - 9) · z^(8 - 9) = z^-23 · z^-1
Finally, we can simplify this further by applying the rule of adding exponents:
z^-23 · z^-1 = z^(-23 - 1) = z^-24
Therefore, the fully simplified expression is z^-24 .
To fully simplify the expression, let's break it down step by step.
First, let's simplify the numerator, (z^-7·z^4)^2. To raise a power to another power, we multiply the exponents. In this case, we have:
(z^-7·z^4)^2 = z^(-7·2)·z^(4·2) = z^-14·z^8
Next, let's simplify the denominator, (z^3·z^3)^3. To multiply exponents with the same base, we add the exponents. Here, we have:
(z^3·z^3)^3 = z^(3+3)·3 = z^6·3 = z^18
Now, let's rewrite the expression with the simplified numerator and denominator:
(z^-7·z^4)^2/z^3·z^3)^3 = (z^-14·z^8)/(z^18)
Since we have a division with the same base, z, we can subtract the exponents:
z^(-14-18)·z^(8) = z^-32·z^8
Finally, to fully simplify the expression, we can combine the terms with the same base:
z^-32·z^8 = z^(-32+8) = z^-24
Therefore, the fully simplified expression is z^-24.