Use a calculator to approximate each irrational number to...

Use a calculator to approximate each irrational number to the nearest hundredth. If the expression is not a real number, so state.
a) $\sqrt[3]{-37}$
b) $-\sqrt[3]{43}$
c) $-\sqrt[3]{-61}$
d) $\sqrt[3]{67}$

Key Concepts

Irrational Numbers

Irrational numbers are numbers that cannot be exactly represented as a ratio of two integers, meaning their decimal representations are non-terminating and non-repeating. In many practical contexts, such as when using a calculator, these numbers must be approximated to a particular degree of precision, like the nearest hundredth.

Cube Roots

A cube root of a number is the value which, when multiplied by itself three times, equals the original number. Cube roots are defined for both positive and negative numbers, since the cube of a negative number is negative, which means that even expressions with negative radicands may yield real number answers.

Calculator Approximations

Calculator approximations involve using technology to determine a numerical estimate for an expression, particularly when the exact value cannot be easily expressed in a simple fraction. Rounding the result to the desired precision, such as to the nearest hundredth, is a common practice in these calculations.

Real Number System

The real number system encompasses all numbers that can be found on the number line, including both rational and irrational numbers. Understanding this system is crucial when determining if an expression yields a real number and how to deal with expressions that might involve complex components, such as in cases where square roots or cube roots of negative numbers are encountered.

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