Evaluate the expression. (a^3)(b^-2)(c^10)(d^15)

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kingattaskus12's profile pic

kingattaskus12 | (Level 3) Adjunct Educator

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Remove the negative exponent in the numerator by rewriting `b^-2`  as `1/b^2`  A negative exponent follows the rule: `a^(-n) = 1/a^n`

`(a^3)(1/b^2)(c^10)(d^15)`


Multiply `a^3`  by `1/b^2`  to get `a^3/b^2`

`(a^3/b^2)(c^10)(d^15)`


Multiply `a^3/b^2`  by `c^10`  to get `(a^3c^10)/b^2`

`((a^3c^10)/b^2)(d^15)`


Multiply `(a^3 c^10)/b^2`  by `d^15`  to get `(a^3 c^10 d^15)/b^2`

`((a^3c^10 d^15)/b^2)`


Remove the parentheses around the expression `(a^2 c^10 d^15)/b^2`

 Therefore, the answer will be `(a^2 c^10 d^15)/b^2`

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should make the supposition that the bases are all different, since the problem does not provides any information regarding them.

You need to use the negative power property for the factor `b^(-2)` , such that:

`b^(-2) = 1/b^2`

You also need to use the following exponent rules, such that:

`(a*b)^x = a^x*b^x`

`a^(x+y) = a^x*a^y`

Hence, reasoning by analogy, you may write the factors `c^10` and `d^15` , such that:

`c^10 = c^(3+7) = c^3*c^7`

`d^15 = d^(3+7+5) = d^3*d^7*d^5`

Hence, you may group the powers in the given expression, such that:

`a^3*c^10*d^15 = (a^3*c^3*d^3)*(c^7*d^7)*d^5`

`a^3*c^10*d^15 = (a*c*d)^3*(c*d)^7*d^5`

Replacing `(a*c*d)^3*(c*d)^7*d^5` for `a^3*c^10*d^15`   in the given expression, yields:

`a^3*b^(-2)*c^10*d^15 = ((a*c*d)^3*(c*d)^7*d^5)/b^2`

Hence, evaluating the given expression, using the exponents rules, under the given conditions, yields `a^3*b^(-2)*c^10*d^15 = ((a*c*d)^3*(c*d)^7*d^5)/b^2.`

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