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) $\sqrt{-101}$
b) $-\sqrt{101}$
c) $\sqrt[3]{-101}$
d) $-\sqrt[3]{-101}$

Key Concepts

Real vs Complex Numbers

This concept distinguishes between numbers that can be found on the real number line and those that involve the imaginary unit i. In the context of square roots, while the square root of a non-negative number is a real number, the square root of a negative number is not defined within the real numbers and is instead expressed in terms of complex numbers.

Square Roots

The square root function finds a number which, when multiplied by itself, gives the original number (the radicand). In the realm of real numbers, taking the square root is only possible when the radicand is zero or positive. If the radicand is negative, the expression results in a complex number.

Cube Roots

Unlike square roots, cube roots are defined for all real numbers because raising any number to the power of three retains the sign of the original number. Therefore, the cube root of a negative number is also negative, making cube roots a broadly applicable operation in the set of real numbers.

Irrational Numbers

Irrational numbers are those that cannot be expressed exactly as a fraction of integers and typically have non-terminating, non-repeating decimal expansions. When calculating approximations, these numbers are rounded to a specified degree of accuracy, such as to the nearest hundredth.

Approximation and Rounding

This concept involves calculating a numerical value that is close enough to the true value of an expression for practical purposes. Rounding is the process of adjusting the number to a given level of precision—in this case, approximating to the nearest hundredth.

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