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The power rule is a fundamental concept in algebra which helps in simplifying expressions with exponents. It consists of two main rules:

• Product of Powers: When multiplying two exponents with the same base, you add their exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When an exponent is raised to another exponent, you multiply their exponents: \((a^m)^n = a^{m \times n}\).

These rules simplify expressions efficiently and reduce the number of steps required. For instance, in the original exercise, you start with the expression inside the parenthesis \(x^4 \times x^3\). Applying the product of powers rule gives us \(x^{4+3} = x^7\).
Using these rules can make complex algebraic expressions a lot simpler and manageable.

Exponentiation is a mathematical operation where a number, called the base, is raised to the power of an exponent. It is written as \(a^n\), where 'a' is the base and 'n' is the exponent. The exponent indicates how many times the base is multiplied by itself.
For example:

• \(2^3 = 2 \times 2 \times 2 = 8\)
• \(5^2 = 5 \times 5 = 25\)

In our exercise, the last step involves exponentiation. We have \((x^7)^2\). According to the power of a power rule, you multiply the exponents: \(7 \times 2\). Therefore, \((x^7)^2 = x^{14}\). Exponentiation greatly simplifies handling large numbers and algebraic expressions.

Algebraic simplification involves reducing expressions to simpler forms without changing their value. It makes equations more manageable and easier to solve. The key steps in simplification usually include:

• Combining like terms
• Using distributive properties
• Applying algebraic identities and rules like the power rule

In the given exercise, simplification started inside the parentheses by combining the powers of 'x': \(x^4 \times x^3\) using the rule \(a^m \times a^n = a^{m+n}\). We got \(x^7\).
Next, to further simplify, we applied the power rule by raising it to the power of 2: \((x^7)^2\), resulting in \(x^{14}\). These techniques turn complicated expressions into simpler ones which are easier to understand and work with.
Remember, practice and familiarity with these rules make simplification quicker and more intuitive.