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Exponent rules are essential for simplifying expressions involving powers. In math, an exponent refers to the number of times a number (the base) is multiplied by itself. Here are some key rules:

When simplifying expressions with negative exponents, remember that a negative exponent means the reciprocal of the base raised to the opposite positive power. For example, \(a^{-n} = \frac{1}{a^n}\).

If you have a product raised to an exponent, distribute the exponent to each factor in the product, i.e., \((ab)^n = a^n \cdot b^n\).

Applying these rules, we can see that for the second part of the problem: \((4 \cdot 10)^{-3}\) is the same as \ \frac{1}{40^3} = 0.000015625\.

Practice and familiarity with these rules will make simplification much easier and quicker.

Scientific notation is a method to express very large or very small numbers compactly. It uses powers of 10. For example, instead of writing 1000, we can write \(10^3\). Likewise, instead of writing 0.001, we can write \(10^{-3}\).

Here are the basics:

• A number in scientific notation is written as the product of a number (between 1 and 10) and a power of 10, like this: \a \times 10^n\.
• The exponent (n) indicates how many places the decimal point has been moved.
• If the exponent is positive, the decimal moves to the right.
• If the exponent is negative, the decimal moves to the left.

In problem part (a), \(4 \times 10^{-3}\) simplifies to \ \frac{4}{1000} = 0.004\. This shows the practical use of scientific notation for simplifying very small numbers.

A strong foundation in basic arithmetic is crucial as it underpins almost all areas of mathematics. Here are some fundamental skills:

• Multiplication: The operation of binary multiplicative process, like \(4 \times 10 = 40\).
• Division: Splitting a number into equal parts, like \(40 \div 64000\).
• Fractions: Understanding how to simplify, for example, \(4 \times 10^{-3} = \frac{4}{1000} = 0.004\).
• Converting between forms: Moving easily between fraction, decimal, and scientific notation.

Breaking numbers carefully and performing operations step by step will help you solve complex problems more accurately. In our exercise, converting the expression to different forms (fraction, decimal) enables simplification and better understanding.