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When working with exponents, one of the fundamental rules is the law for multiply exponents with the same base. To multiply expressions with the same base, you just need to add the exponents together. This is incredibly useful for simplifying expressions.

Let's consider the expression \(u^{2} \times u^{3} \times u\). Here \(u\) is the common base for each term. According to our rule, we add the exponents: \(2 + 3 + 1 = 6\). This gives us the simplified expression \(u^{6}\).

Understanding the base and exponent is crucial in working with exponential expressions. The base is the number that is being multiplied by itself, and the exponent tells us how many times the base is used as a multiplier. For example, in \(u^{2}\), \(u\) is the base, and the exponent is 2, meaning that \(u\) is used as a multiplier twice: \(u \times u\).

Every number in an exponential expression has these two parts. Even when an exponent isn't visibly written, such as in \(u\), it is understood to be \(u^{1}\) because the base \(u\) is used once as the multiplier.

The process of simplifying expressions with exponents involves using rules like the product of powers rule which we explored in previous sections. Another important rule is the power of a power rule, which says that to raise an exponent to another power, you multiply the exponents.

Additionally, if an expression contains a constant multiplied by a base with an exponent, you simplify by maintaining the constant and applying the exponent rule to the base. For instance, in the expression \(5u^{2} \times 3u^{3}\), we first multiply the constants (\(5 \times 3 = 15\)) and then apply the exponent rule for \(u^{2} \times u^{3}\) we saw earlier to get \(15u^{5}\). Understanding and applying these rules correctly results in a simplified and more manageable expression.