Each row of the table contains two exponential expressions. For each row, determine whether Expression 1 is equivalent to Expression 2 for all values of x and y where the expressions are defined.
determine if they are equivalent or not by saying " yes " or " no "
Expression 1 : (2x^2)^4 Expression 2: 8x^8 are they both equivalent?
Expression 1: 4x^-1 Expression 2: 4/x are they both equivalent?
Expression 1: (x/y^2)^2 Expression 2: y^4/x^2 are they both equivalent?
Expression 1: *5x^5)^0 Expression 2: 0 are they both equivalent?
For each pair of expressions, let's consider whether they are equivalent.
1. Expression 1: (2x^2)^4 ; Expression 2: 8x^8
When we simplify Expression 1, we'll raise both the base and the exponent inside the parenthesis to the power of 4:
(2x^2)^4 = 2^4 * (x^2)^4 = 16 * x^8
Since 16 is not equal to 8, Expression 1 is not the same as Expression 2. Thus the answer is "no".
2. Expression 1: 4x^-1 ; Expression 2: 4/x
Expression 1 can be rewritten as a fraction where x^-1 is equivalent to 1/x:
4x^-1= 4 / x
Since this is the same as Expression 2, the two expressions are equivalent. Thus the answer is "yes".
3. Expression 1: (x/y^2)^2 ; Expression 2: y^4/x^2
First we'll simplify Expression 1 by squaring both the numerator and the denominator:
(x/y^2)^2 = x^2 / (y^2)^2 = x^2 / y^4
This is the reciprocal of Expression 2, not equivalent to it. Thus the answer is "no".
4. Expression 1: (5x^5)^0 ; Expression 2: 0
Any non-zero number raised to the power of 0 is equal to 1, so Expression 1 simplifies to:
(5x^5)^0 = 1 (as long as x is not zero, because if x was zero, there would be a point of indeterminacy)
Expression 2 is simply 0. Since 1 is not equal to 0, the expressions are not equivalent. Thus the answer is "no".