simplify and write using only positive exponents. thanks.
(2x^3y^4)^-2 divided by (8xy^6)^-3
[ 8^3 x^3 y^18 ] / [2^2 x^6 y^8 ]
[ 2^9 / 2^2 ] [ x^-3] [ y^10 ]
[ 2^7 ] y^10/x^3
128 y^10/x^3
To simplify and write the given expression using only positive exponents, we can follow these steps:
Step 1: Reciprocal of the expression
Take the reciprocal of the given expression to change the negative exponents to positive exponents. This is done by flipping the expression:
(2x^3y^4)^-2 divided by (8xy^6)^-3
becomes
(1 / (2x^3y^4)^2) divided by (1 / (8xy^6)^3)
Step 2: Apply the property of exponents
By applying the property of exponents, we can simplify the expression. According to the property, when we raise a power to another power, we multiply the exponents:
(1 / (2x^3y^4)^2) divided by (1 / (8xy^6)^3)
becomes
(1 / (2^2 * x^(3*2) * y^(4*2))) divided by (1 / (8^3 * x * y^(6*3)))
Step 3: Simplify further
Calculate the exponents and simplify the expression:
(1 / (4 * x^6 * y^8)) divided by (1 / (512 * x * y^18))
becomes
(1 * (512 * x * y^18)) / (4 * x^6 * y^8)
which simplifies to
(512xy^18) / (4x^6y^8)
Step 4: Further simplify by canceling out common factors
In this step, we cancel out the common factors between the numerator and denominator:
(512xy^18) / (4x^6y^8)
= (512 / 4) * (x / x^6) * (y^18 / y^8)
= 128 * (1 / x^5) * y^(18-8)
= 128 / x^5 * y^10
So, the simplified expression with only positive exponents is:
128 / x^5 * y^10