simplify using the exponent law. express your answer using positive exponents only.
(x^2)^3(2x^3)^-3/(4x)^2
( x ^ 2 ) ^ 3 = x ^ 6
( 2 x ^ 3 ) ^ ( - 3 ) = 2 ^ ( - 3 ) * x ^ [ ( - 3 ) * ( - 3 ) ] =
2 ^ ( - 3 ) * x ^ ( - 9 )
1 / ( 2 ^ 3 * x ^ 9 ) = 1 / 8 x ^ 9
3 /( 4 x )^ 2 = 3 / ( 4 ^ 2 x ^ 2 ) = 1 / 16 x ^ 2
( x ^ 2 )^ 3 ( 2 x ^ 3 )^ -3 / ( 4 x ) ^ 2 =
x ^ 6 * 1 / 8 x ^ 9 * 1 / 16 x ^ 2 =
x ^ 6 / ( 8 x ^ 9 * 16 x ^ 2 ) =
x ^ 6 / ( 128 * x ^ 11 ) =
x ^ 6 / ( 128 * x ^ 6 * x ^ 5 ) =
1 / 128 x ^ 5
CORRECTION:
( 2 x ^ 3 ) ^ ( - 3 ) = 2 ^ ( - 3 ) * x ^ [ 3 * ( - 3 ) ] =
2 ^ ( - 3 ) * x ^ ( - 9 )
1 / ( 2 ^ 3 * x ^ 9 ) = 1 / 8 x ^ 9
To simplify the expression (x^2)^3(2x^3)^-3/(4x)^2 using the exponent law, we can follow these steps:
Step 1: Apply the power of a power rule.
(x^2)^3 = x^(2*3) = x^6
Step 2: Apply the power of a power rule for the second term.
(2x^3)^-3 = (2^-3)(x^3)^-3 = (1/2^3)(x^(3*-3)) = (1/8)(x^-9)
Step 3: Simplify the denominator using the power of a product rule.
(4x)^2 = 4^2(x^2) = 16x^2
Now, we can substitute these simplified expressions back into the original expression:
(x^2)^3(2x^3)^-3/(4x)^2 = (x^6)(1/8x^-9)/(16x^2)
Step 4: Combine the numeric factors.
= (1/8)(1/16)(x^6)(x^-9)/(x^2)
Step 5: Combine the x terms using the quotient rule for exponents.
= (1/128)(x^(6-9))/(x^2)
= (1/128)(x^-3)/(x^2)
= (1/128)(1/x^3)/(x^2)
Step 6: Combine the fractions by multiplying the numerator and denominator by x^3.
= (1/128)(1)/(x^3)(x^2)
= 1/(128x^5)
Therefore, the simplified expression is 1/(128x^5) using positive exponents only.