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Evaluating expressions is an essential skill in mathematics that involves simplifying mathematical phrases. In algebra, this typically means finding the value of a given expression by substituting variables with numbers or by simplifying the expression itself.

Let's look at the expression given in our exercise: \[x^{0} + (2x)^{0}\] To evaluate this, you need to simplify each term separately using known mathematical rules, such as the zero exponent rule, before performing any operations like addition or subtraction. This ensures clarity and precision in solving algebraic problems.

In this particular task, the first step is working with exponents, which brings us to our next concept: the zero exponent rule.

The zero exponent rule is one of the fundamental rules of exponents. It states that any nonzero base raised to the power of zero is equal to 1. This might seem a little surprising at first but is quite logical when you consider the pattern of decreasing exponents.

• For example, when you have \(x^1 = x\), then \(x^0\) is simply the previous term divided by \(x\), resulting in \(1\).
• So, \(x^0 = 1\) for any non-zero \(x\).

Let's apply this rule to our original expression:
\(x^{0} + (2x)^{0}\)
Using the zero exponent rule, both \(x^0\) and \((2x)^0\) simplify to 1, giving:

• \(x^0 = 1\)
• \((2x)^0 = 1\)

Understanding and using this rule simplifies your expressions easily and is key to mastering exponent problems in algebra.

Algebraic simplification involves rewriting expressions in the most compact or efficient form without changing their value. By applying basic algebraic rules, you can reduce expressions to simpler forms, which are easier to work with or evaluate. In this exercise, after applying the zero exponent rule, we have:
\(x^{0} + (2x)^{0} = 1 + 1\)
Now, our task is straightforward: add the numbers together.

• We perform simple addition: \(1 + 1 = 2\).

This outcome represents the simplest form of the original problem, showing how different algebraic steps can reduce complex expressions.

Mastering algebraic simplification by systematically applying rules and performing operations helps solve problems, making mathematical tasks more manageable and intuitive.