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Understanding how to compare expressions with exponents is a fundamental skill in mathematics. When expressions have the same base, such as the number 4 in this example, we can simply compare their exponents to determine the larger value.

In our exercise, we are asked to compare two expressions: \(4^{2} \cdot 4^{8}\) and \(4^{16}\). Both expressions have the same base (4), making our job easier. In essence, comparison of exponents equates to comparing their powers when the base remains constant. The expression with the greater exponent will invariably represent the larger quantity. It is akin to stacking layers - the more layers (greater exponent), the higher the pile (larger the value).

After simplifying the left side, \(4^{2} \cdot 4^{8} = 4^{10}\), we end up with two expressions: \(4^{10}\) and \(4^{16}\). With this, a direct comparison reveals that 10 is less than 16, which tells us \(4^{10} < 4^{16}\). Simple comparisons like these underscore the central rule that for a given positive base greater than one, the expression with the larger exponent will always be greater.

Mastering exponent rules streamlines the process of working with exponential expressions in mathematics. The exercise presented provides a splendid instance to apply one of the most essential rules - the Product of Powers.

Product of Powers

This rule states that when multiplying two exponents with the same base, you add the exponents. Mathematically, it is expressed as \( a^{m} \cdot a^{n} = a^{m+n} \).

Therefore, when we deal with \( 4^{2} \cdot 4^{8} \), we sum the exponents 2 and 8, which gives us \( 4^{2+8} = 4^{10} \). It's a critical tool, simplifying what could be an arduous multiplication of long strings of the same number. In this manner, exponent rules not only make calculations more efficient but provide clarity in understanding the relative magnitude of exponential expressions.

Mathematical inequalities are expressions that define the relative size or order of two values. They are represented by symbols like \(>\) for 'greater than' and \(<\) for 'less than'. The employment of inequalities is prevalent in various branches of mathematics and real-life situations where precise amounts might be unknown, but a comparison can be established.

Solving the given exercise involves determining the correct inequality symbol to place between \(4^{10}\) and \(4^{16}\). As previously established, inputting the correct comparison is made easier since we have simplified our expressions using exponent rules. Recognizing that 10 is smaller than 16 leads us to conclude that \(4^{10}\) is less than \(4^{16}\), hence \(4^{10} < 4^{16}\).

It's essential to become comfortable with inequalities because they are not just confined to isolated equations but also play a pivotal role in graphing solutions and understanding functions within the coordinate plane.