9514 1404 393
Answer:
100
Step-by-step explanation:
A root is equivalent to a fractional power. A square root is a 1/2 power. A cube root is a 1/3 power. The index of the root is the denominator of the fraction.
Of course the usual rules of exponents apply in evaluating an expression.
(a^b)^c = a^(bc)
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Your expression can be simplified as follows.
[tex]\displaystyle\sqrt[3]{1000^2}=(1000^2)^{\frac{1}{3}}=1000^{\frac{2}{3}}=(10^3)^{\frac{2}{3}}=10^2=\boxed{100}[/tex]
It can also be simplified another way:
[tex]\displaystyle\sqrt[3]{1000^2}=\sqrt[3]{(10^3)^2}=\sqrt[3]{(10^2)^3}=10^2=100[/tex]
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Additional comments
A problem like this is simpler if you are familiar with the squares and cubes of small integers.
