Provide two different ways to simplify each of the problems below. Please write all final answers using positive exponents.
(-4m^-2 n^2)^3 * (2m^3 n)^-1
method 1: method 2:
(X^-1/2 Y^3)^-2
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XY^1/3 * X^2 Y^-4
(-4)^3 m^-6 n^6 2^-1 m^-3 n^-1
(-64) m^-9 n^5/2
-32 n^5/m^9
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(-4m^-2 n^2)^3 / (2m^3 n)
-1^3 * 2^6 * n^6 / [m^6 * n * 2 m^3]
-1 * 2^5 * n^5 /m^9
-32 n^5/m^9
To simplify the given expressions, we can use two different methods. Let's break down each method and solve the problems step by step.
Method 1:
For the expression (-4m^-2 n^2)^3 * (2m^3 n)^-1:
Step 1: Simplify inside the parentheses.
-4m^-2 can be written as -4/m^2, and (2m^3 n)^-1 can be written as 1/(2m^3 n).
Step 2: Apply the exponent rule for raising a power to a power.
(-4/m^2)^3 = -64/m^6
Step 3: Multiply the expressions.
(-64/m^6) * (1/(2m^3 n)) = -64/(2m^9 n)
Step 4: Simplify further.
-64/(2m^9 n) can be reduced by dividing the numerator and the denominator by 2.
-32/(m^9 n)
Method 2:
For the expression (X^-1/2 Y^3)^-2 / (XY^1/3 * X^2 Y^-4):
Step 1: Deal with exponents inside the parentheses first.
(X^-1/2 Y^3)^-2 can be written as 1/(X^1 Y^-6).
Step 2: Simplify the denominator.
XY^1/3 * X^2 Y^-4 can be written as X^(1+2) * Y^(1/3 + (-4)), which simplifies to X^3 * Y^(-11/3).
Step 3: Apply the exponent rule for dividing exponents.
1 / (X^1 Y^-6) divided by (X^3 * Y^(-11/3)) is equal to (1 * X^-3 * Y^(6 + 11/3)).
Step 4: Simplify the resulting expression.
Multiply the coefficients and add the exponents: X^-3 * Y^(6 + 11/3) = X^-3 * Y^(18/3 + 11/3) = X^-3 * Y^(29/3).
Therefore, the final answers using positive exponents for the two problems are:
Method 1: -32/(m^9 n)
Method 2: X^-3 * Y^(29/3)