Simplify (y^9)^5\cdoty^-6
To simplify the expression, we use the power of a power rule, which states that (a^m)^n = a^(m*n).
In this case, we have (y^9)^5 * y^-6.
By applying the power of a power rule, we multiply the exponents: 9 * 5 = 45.
Therefore, the expression simplifies to y^45 * y^-6.
To combine the terms with the same base, we use the product of powers rule, which states that a^m * a^n = a^(m + n).
In this case, we have y^45 * y^-6.
By applying the product of powers rule, we add the exponents: 45 + (-6) = 39.
Therefore, the final simplified expression is y^39.
Simplify (z^5)^4×(z^5)^2
To simplify this expression, we first apply the power of a power rule, which states that (a^m)^n = a^(m*n).
In this case, we have (z^5)^4 * (z^5)^2.
By applying the power of a power rule, we multiply the exponents: 5*4 = 20 for the first term, and 5*2 = 10 for the second term.
Therefore, the expression simplifies to z^20 * z^10.
Next, we use the product of powers rule, which states that a^m * a^n = a^(m + n).
In this case, we have z^20 * z^10.
By applying the product of powers rule, we add the exponents: 20 + 10 = 30.
Therefore, the final simplified expression is z^30.
Simplify (z^-8\cdotz^5)^3/(z^2\cdotz^2)^3
To simplify this expression, let's simplify the numerator and the denominator separately, and then divide the two results.
Numerator:
z^(-8) * z^5 = z^(-8 + 5) = z^(-3)
Denominator:
(z^2 * z^2)^3 = z^(2*3) * z^(2*3) = z^6 * z^6 = z^(6 + 6) = z^12
Now, let's divide the numerator by the denominator:
(z^(-3)) / (z^12) = z^(-3 - 12) = z^(-15)
Therefore, the simplified expression is z^(-15).
Simplify z^-8\cdotz^5)^3/z^2\cdotz^2)^3
To simplify this expression, we need to use the properties of exponents.
First, let's simplify the numerator: (z^-8 * z^5)^3
Let's combine the exponents inside the parentheses: -8 * 3 = -24. So, the numerator simplifies to z^-24.
Now, let's simplify the denominator: (z^2 * z^2)^3
Let's combine the exponents inside the parentheses: 2 * 3 = 6. So, the denominator simplifies to (z^6)^3 = z^18.
Now that we have both the numerator and denominator simplified, we can write the expression as:
z^-24 / z^18
To divide exponential expressions with the same base, we subtract the exponents:
z^-24 / z^18 = z^(-24-18) = z^-42.
Therefore, the simplified expression is z^-42.