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Exponents are a way to express repeated multiplication of a number by itself. For example, 2 raised to the power of 3 is written as \(2^3\) and means 2 × 2 × 2, which equals 8. Similarly, \(10^3\) means 10 × 10 × 10, which equals 1000. When dealing with exponents, there are specific rules that simplify calculations:

• Product of Powers Rule: When multiplying two powers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{m \times n}\).
• Negative Exponents: A negative exponent means the reciprocal of the base raised to the opposite positive exponent: \(a^{-n} = \frac{1}{a^n}\).

These rules help simplify calculations involving large or small numbers, especially when expressed in scientific notation.

Multiplying powers can look complex, but it's straightforward if you use the right rules. Let's break it down with our example:
We have \((2 \times 10^{-9})^3\) and \((5 \times 10^3)^2\).
First, take \({2 \times 10^{-9}}\) and raise it to the power of 3. This can be separated as: \(2^3 \times (10^{-9})^3\). Calculate \(2^3 = 8\) and use the rule of exponents for the power of 10: \((10^{-9})^3 = 10^{-27}\). Therefore: \((2 \times 10^{-9})^3 = 8 \times 10^{-27}\).
Next, take \(5 \times 10^3\) and raise it to the power of 2. This can be separated as: \(5^2 \times (10^3)^2\). Calculate \(5^2 = 25\) and apply the exponent rule for 10: \((10^3)^2 = 10^6\). Therefore: \((5 \times 10^3)^2 = 25 \times 10^6\).
Now, multiply the results from both parts: \(8 \times 10^{-27} \times 25 \times 10^6\). Combine the coefficients \(8 \times 25 = 200\) and combine the powers of \(10\): \(10^{-27} \times 10^6 = 10^{-21}\). Thus, the result is \(200 \times 10^{-21}\).

Scientific notation is a way to express extremely large or small numbers in a compact form. A number in scientific notation is written as \(a \times 10^n\), where \(a\) is a number between 1 and 10, and \(n\) is an integer. This notation makes it easier to read, understand, and work with these numbers.
In our example, we have \(200 \times 10^{-21}\). To convert this into proper scientific notation, let's follow these steps:
First, rewrite \(200\) as \(2.0 \times 10^2\). Now our expression becomes \(2.0 \times 10^2 \times 10^{-21}\).
Next, combine the powers of \(10\): \(10^2 \times 10^{-21}\) by adding the exponents: \(2 + (-21) = -19\). Therefore, \(10^2 \times 10^{-21} = 10^{-19}\).
So, the final expression in scientific notation is \(2.0 \times 10^{-19}\). This method keeps calculations straightforward and ensures your results are consistent and easy to interpret.