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Simplifying expressions in algebra involves breaking down a complex equation into its simplest form. This process often makes it easier to work with the expression in subsequent calculations. Simplifying can involve several steps depending on the complexity of the expression. For instance, you might need to combine like terms, factor the expression, or perform arithmetic on the coefficients.

In our example, given the expression \((3 x)(5 x)(4 x)\), simplifying it involves multiplying both the coefficients and the variables step-by-step. This reduces the expression to a form that is more concise and easier to handle in further operations.

One key step in simplifying expressions is multiplying the coefficients. Coefficients are the numerical values that accompany variables in an algebraic term.

In our expression \((3 x)(5 x)(4 x)\), the coefficients are 3, 5, and 4. To multiply them, you simply perform the arithmetic operation: \[ 3 \times 5 \times 4 = 60 \]
This result gives us the new coefficient for our simplified expression.

Remember that coefficients multiply together just like any ordinary numbers. Hence, even if the coefficients were negative or fractions, the same rules apply. This step often significantly reduces the complexity of the expression.

Exponents are a way to denote repeated multiplication of a number or variable. For example, \(x^3\) means \(x \times x \times x\). When dealing with exponents, there are specific rules to follow.

In our exercise, each term contains the variable \(x\) with an implied exponent of 1: \(x^1\). When multiplying variables with the same base, you add their exponents. Therefore: \[ x^1 \times x^1 \times x^1 = x^{1+1+1} = x^3 \]
This rule simplifies the variables into a single term with a new exponent, streamlining the overall expression.

It's crucial to clearly understand how exponents work because they frequently appear in algebraic expressions. Misunderstanding how to handle them can lead to errors in simplification and further calculations.