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Exponents are used to represent repeated multiplication of a number by itself. When you see a number like \(a^n\), it means that you multiply \(a\) by itself \(n\) times. For example, \(2^3 = 2 \times 2 \times 2 = 8\).

The base \(a\) is the number that gets multiplied, and \(n\), called the exponent, tells us how many times the base appears in the multiplication.

When simplifying expressions with exponents, carefully follow the rules and apply them to each part of the term. In our example of simplifying \((-5x)^4\), both the number and the variable have to be raised to the power of 4.

This means we apply the exponent to \(-5\) and \(x\) separately.

Remember that any non-zero number raised to the power of 0 is 1. For instance, \(7^0 = 1\) and \((-2x)^0 = 1\). This is because multiplying a number by itself zero times leaves you with the multiplicative identity, which is 1.

A coefficient is a number that multiplies a variable. In the term \(-5x\), \(-5\) is the coefficient. When dealing with coefficients, we need to understand how to handle both positive and negative numbers.

When you raise a coefficient to a power, you apply the exponent only to the number itself. For example, when we simplify \((-5)^4\), we use the rule of negative exponents: \((-a)^n = (-1)^n \times a^n\). Here, because \((-1)^4 = 1\), we simply calculate \(-5^4\) as \(5^4 = 625\). So \((-5)^4 = 625\).

This rule helps us understand why the product in this example is positive. The even exponent neutralizes the negative sign, leaving us with a positive outcome.

Power rules help us simplify expressions where terms are raised to exponents. These rules show us how to manage various cases of exponentiation.

Some important power rules include:

• \((a^m)^n = a^{mn}\)
• \frac{a^m}{a^n} = a^{m-n}\ )
• \