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Number systems are the foundation of numerical representation in mathematics. The most common number system that we use is the decimal system, which is base 10. In this system, digits range from 0 to 9, and each position represents a power of 10. However, many other number systems exist. For example, binary is base 2, and represents numbers using only 0s and 1s.
In the problem we're examining, the number system base is expressed as \(b\). To understand and convert numbers between different bases, it's important to know that each digit represents a power of the base. For a number in base \(b\), the digits represent powers starting from \(b^0\) on the rightmost digit, increasing to \(b^1, b^2,\) and so on as we move leftwards.
Understanding these concepts allows us to convert and manipulate numbers across various bases, an important skill in many areas of math and computer science.

Powers of numbers are a critical concept in mathematics, involving repeated multiplication of a number by itself. When a number is raised to a power, it is known as an "exponent."
For instance, in \(b^3\), the number \(b\) is multiplied by itself twice more (totaling three instances of \(b\)). This can be expanded as \(b \times b \times b\).
When dealing with number systems, powers are used to determine the weight of each digit's position in the number. This is why, for the number \(1001_b\), each digit is multiplied by a power of \(b\). The expression \(1 \cdot b^3 + 0 \cdot b^2 + 0 \cdot b^1 + 1 \cdot b^0\) demonstrates this, where each term represents the digit multiplied by the corresponding power of the base.
The understanding of powers aids in simplifying and solving equations, particularly in conversions between various number systems.

The cube root of a number is a value that, when multiplied by itself three times, yields the original number. It's the opposite operation of cubing a number. For example, the cube root of \(8\) is \(2\) because \(2 \times 2 \times 2 = 8\).
In solving equations, finding the cube root can isolate the variable, as seen in our example where \(b^3 = 125\). To find \(b\), we need to determine what number cubed gives us \(125\). Computing \(\sqrt[3]{125} \), we find that \(5 \times 5 \times 5 = 125\), so \(b = 5\).
Cube roots are particularly useful in solving polynomial equations, especially where cubes or higher powers are involved. They offer an essential approach in algebra for isolating variables and finding precise solutions.