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Exponents are a convenient way to represent repeated multiplication of the same number by itself. To truly master exponent rules, a few key guidelines are essential:

• Product of Powers Rule: When multiplying like bases, you add the exponents, such as \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When raising a power to another power, you multiply the exponents, like \((a^m)^n = a^{m \times n}\).
• Power of a Product Rule: When raising a product to a power, you apply the exponent to each factor in the product, such as \((ab)^m = a^m \times b^m\).
• Quotient of Powers Rule: When dividing like bases, you subtract the exponents, as in \(a^m / a^n = a^{m-n}\).

To understand how these rules apply, let's consider the given exercise. In step 1, the expression \((8 \times 10^{-5})^3\) needs to be simplified. According to the Power of a Product Rule, we raise both 8 and \(10^{-5}\) to the power of 3.

Scientific notation is a way of expressing very large or very small numbers in a compact form. It is particularly useful in scientific fields where measurements can span many orders of magnitude. A number in scientific notation is written as the product of a coefficient and a power of ten:

\text{Number} = \text{Coefficient} \times 10^{\text{Exponent}}\

The coefficient must be a number between 1 and 10, and the exponent is an integer. For example, the number 5000 can be written in scientific notation as \5 \times 10^3\. Similarly, the number 0.0008 can be written as \8 \times 10^{-4}\.

In the given exercise, we begin with the expression \(8 \times 10^{-5}\). This is already in scientific notation, where 8 is the coefficient and \10^{-5}\ is the power of ten. When we cube this expression, each part is cubed separately according to the rules of exponents.

Cubing a number means raising that number to the power of three. Essentially, you multiply the number by itself three times. Mathematically, cubing a number \(a\) is written as \(a^3\).

For example:
\ 2^3 = 2 \times 2 \times 2 = 8\
\ 7^3 = 7 \times 7 \times 7 = 343\

In our exercise, we are required to cube both the coefficient 8 and the power of ten \((10^{-5})\). The calculation of \8^3\ results in 512, which involves multiplying 8 by itself twice:
\ 8 \times 8 = 64\
\ 64 \times 8 = 512\

We then apply the exponent rule to cube the power of ten: \((10^{-5})^3 = 10^{-5 \times 3} = 10^{-15}\). Finally, combining these results gives us \512 \times 10^{-15}\.

Powers of ten are a fundamental concept in mathematics, especially in scientific notation. They provide a way to express very large or very small numbers compactly. A power of ten is written as \10^n\, where \ is an integer.

For example:

• \ 10^3 = 1000 \
• \ 10^{-2} = 0.01 \
• \ 10^0 = 1 \

Negative exponents indicate division by a power of 10, while positive exponents indicate multiplication. In our exercise, the power of ten is initially \10^{-5}\, which means we divide by 100,000 (since \10^{-5} = 1/100000\).

When we cube this power using the exponent rule for powers of a power, \((10^{-5})^3 = 10^{-15}\. This shows how the exponent rules apply even to powers of ten in scientific notation.