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The power rule for exponents is a useful tool when simplifying expressions that involve raising a power to another power.
This rule is stated mathematically as \( \big(a^m\big)^n = a^{m \times n} \). This means you multiply the exponents.
Let's look at an example from the exercise: \big(10 p^{4}\big)^{3}.
Using the power rule, we get: \(10^{3} \times p^{4 \times 3} = 10^{3} \times p^{12}\).
Similarly, for the other term in the expression, \big(5 p^{6}\big)^{2}, the power rule gives us: \(5^{2} \times p^{6 \times 2} = 5^{2} \times p^{12} \). Notice how we effectively distributed the exponent inside the parenthesis to both the numerical coefficient and the variable with its exponent.

Numerical coefficients are the numbers that multiply the variables in an algebraic expression.
In the exercise, the numerical coefficients are 10 and 5. After applying the power rule: \(10^{3} = 1000\) and \(5^{2} = 25\).
When simplifying, you compute these values to make your expression easier to manage: \(1000 p^{12} \times 25 p^{12}\).
It's vital to handle the numerical part separately to avoid confusion. Once you have dealt with the coefficients, you can move to simplify the variables and their exponents.

The product rule for exponents is used when you multiply two expressions with the same base.
The rule states: \(a^m \times a^n = a^{m+n}\). This means you add the exponents.
For the variables in our expression, both parts have the same base, p.
So, we use the product rule: \(p^{12} \times p^{12} = p^{24}\).
Finally, you combine this result with the simplified numerical coefficient: \(25000 p^{24}\).
Remember, this rule applies strictly to terms with the same base and helps in combining like terms when simplifying algebraic expressions.