# What is 100^(lg2)+ (-27)^(1/3)?

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### 2 Answers

You need to rewrite `100` as a power of `10` and `27` as a power of `3` , such that:

`100 = 10^2`

`27 = 3^3`

Replacing `10^2` for `100` and `3^3` for `27` , yields:

`100^(lg2)+ (-27)^(1/3) = (10^2)^(lg 2) + (-3^3)^(1/3)`

You need to use the exponent laws yields:

`(10^2)^(lg 2) + (-3^3)^(1/3) = (10)^(2lg 2) + (-3)^(3*1/3)`

You need to use the power property of logarithms in case of exponent `2lg2` , such that:

`(10)^(2lg 2) + (-3)^(3*1/3) = (10)^(lg 2^2) + (-3)^(3*1/3)`

You need to reduce the duplicate factors in case of the exponent `(3*1/3)` , such that:

`(10)^(lg 2^2) + (-3)^(3*1/3) = (10)^(lg 4) + (-3)^1`

You need to use the following property of logarithms, such that:

`a^(log_a b) = b`

Reasoning by analogy yields:

`(10)^(lg 4) = 4`

`(10)^(lg 4) + (-3)^1 = 4 - 3 = 1`

**Hence, evaluating the given expression, using the exponents laws and the properties of logarithms, yields **`100^(lg2)+ (-27)^(1/3) = 1.`

`100^(log2) = 10^2^(log2) = 10^(2log2) = 10^(log4) = 4`

Exponent/Log rules review: `a^b^c = a^(b*c)`

`10^log(c) = c`

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

``Second part:

`(-27)^1/3 =root(3)(-27)=-3`

(Think, what three numbers multiply to make -27?)

Answer:4 + -3 = **1 **