Properties of Exponents Simplify:  ` (y^x cdot y^(-x))^4`

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tiburtius's profile pic

tiburtius | High School Teacher | (Level 2) Educator

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We will use the following rule:

`a^bcdota^c=a^(b+x)`

Hence you have

`(y^xcdoty^(-x))^4=(y^(x-x))^4=(y^0)^4`

Now since for any `ane0,` `a^0=1` if we assume that `yne0` then you have

`(y^0)^4=1^4=1`  

So if  `yne0` then `(y^xcdoty^(-x))^4=1`  

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kingattaskus12 | (Level 3) Adjunct Educator

Posted on

Remove the negative exponent in the numerator by rewriting `y^(-x)`  as `1/(y^x)`  A negative exponent follows the rule: `a^(-n) = 1/(a^n)`

`(y^x * 1/(y^x))^4`


Multiply `y^x`  by `1/(y^x)`   to get `(y^x)/(y^x)`  

`((y^x)/(y^x))^4`


Reduce the exponents of y by subtracting the denominator exponents from the numerator exponents.

`(y^(x-(x)))^4`


Multiply `-1` by each term inside the parentheses.

`(y^(x-x))^4`


Since x and -x are like terms, add `-x` to `x` to get `0` .

`(y^0)^4`


Expand the exponent `(4)` to the expression.

Therefore, the answer will be `1`

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