Simplify this expression: (2/3)^2 x(6xy^-1)^3
To simplify the expression (2/3)^2 x (6xy^-1)^3, we will follow the order of operations, which is also known as PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, and Addition and Subtraction from left to right).
1. First, simplify the exponent (2/3)^2:
To calculate a fraction raised to a power, we raise both the numerator and denominator to the power. Thus, (2/3)^2 becomes (2^2)/(3^2).
Simplifying further, we obtain 4/9.
Now our expression is: (4/9) x (6xy^-1)^3.
2. Next, let's simplify the exponent (6xy^-1)^3:
When an exponent is applied to a term with multiple factors, we raise each factor to the exponent separately. In this case, we raise 6, x, and y^-1 to the power of 3.
(6^3) x (x^3) x (y^-1)^3.
Simplifying each term, we get 216 x x^3 x (y^-1)^3.
Note that (y^-1)^3 means y raised to the power of -1, cubed.
3. Further simplify the expression:
216 is a constant term and doesn't require any additional simplification.
x^3 is already in its simplest form.
(y^-1)^3 represents y raised to the power of -1, cubed. When we have a negative exponent raised to another exponent, it becomes positive. Therefore, (y^-1)^3 is the same as y^(-1 * 3) = y^-3 = 1/y^3.
Combining all the simplifications, we get:
(4/9) x 216 x x^3 x 1/y^3
Simplifying further,
(4/9) x 216 x x^3/y^3
Multiplying 4/9 by 216 gives us 864/9, which simplifies to 96.
The final simplified expression is:
96 x x^3/y^3