What is X-^2y(x^3y^5)^3 in simplest form for all the values of Z and Y where the expression is defined?
To simplify the expression X^(-2y)(X^3Y^5)^3, we must first apply the exponent properties.
Given expression: X^(-2y) * (X^3Y^5)^3
The expression (X^3Y^5)^3 has two components being raised to the power of 3: X^3 and Y^5. We will raise each to the power of 3 separately.
(X^3)^3 = X^(3*3) = X^9
(Y^5)^3 = Y^(5*3) = Y^15
Bringing these back together, we get:
(X^3Y^5)^3 = X^9 * Y^15
Now, let's combine this with X^(-2y):
X^(-2y) * (X^9 * Y^15)
When multiplying expressions with the same base, we add the exponents:
X^(-2y + 9) * Y^15
This is the simplified form of the given expression, for all values of X and Y where X is nonzero (since you cannot raise 0 to a negative exponent), and Y can be any real number (since Y is only being raised to positive exponents). Note that there's a critical assumption that X or Y is not zero since division by zero or raising zero to a negative power would be undefined.
I only have a box for X and Y
Apologies for the confusion. Let's break down the expression step by step using the correct variables X and Y where applicable.
The expression is X^(-2Y) * (X^3Y^5)^3.
To simplify the expression:
1. Expand the power of the term (X^3Y^5)^3 by applying the power to each component inside the parentheses.
(X^3)^3 * (Y^5)^3 = X^(3*3) * Y^(5*3) = X^9 * Y^15
2. Now, the original expression is:
X^(-2Y) * X^9 * Y^15
3. We can then merge the X terms by adding their exponents since they have the same base:
X^(-2Y + 9)
And then we can rewrite the expression as:
X^(-2Y + 9) * Y^15
This is the expression in its simplest form for all values of X and Y where the expression is defined. Remember, X should not be zero because any non-zero number raised to a power is valid, but for 0, only non-negative exponents are valid. Y is being raised to a positive power, so it has no restrictions regarding its value in this context.
To simplify the expression X^(-2)y(x^3y^5)^3, let's break it down step by step.
Step 1: Simplify the innermost parentheses:
(x^3y^5)^3 = x^(3 * 3) * y^(5 * 3) = x^9 * y^15
Step 2: Substitute the simplified inner expression back into the original expression:
X^(-2)y(x^3y^5)^3 = X^(-2)y(x^9 * y^15)
Step 3: Simplify the expression further using the rules of exponents:
x^(-2) = 1/x^2, so X^(-2)y(x^9 * y^15) = (y/x^2)(x^9 * y^15)
Step 4: Rearrange the terms:
(y/x^2)(x^9 * y^15) = (x^9 * y^15)(y/x^2)
Step 5: Combine like terms:
(x^9 * y^15)(y/x^2) = x^9 * y^(15+1) / x^2 = x^9 * y^16 / x^2 = x^(9-2) * y^16 = x^7 * y^16
Therefore, the simplified form of X^(-2)y(x^3y^5)^3 is x^7 * y^16 for all values of Z and Y where the expression is defined.