91Ó°ÊÓ

In mathematics, an exponent tells us how many times a number, known as the base, is multiplied by itself. When dealing with negative exponents, the calculation follows a specific rule: any number raised to a negative exponent is equivalent to 1 divided by that number raised to the positive exponent. For example, if we have \(x^{-n}\), we can rewrite it as \(\frac{1}{x^n}\).

This rule helps in transforming expressions containing negative exponents into fractions, making them more manageable. For instance, in the original exercise, we had \(a^{-7}\). Using the negative exponent rule, we transformed it to \(\frac{1}{a^7}\). This conversion simplifies calculations and makes it easier to combine terms in further steps.

Exponent rules are fundamental tools for simplifying expressions that involve powers. There are several key exponent rules:

• Product of Powers Rule: \(x^m \cdot x^n = x^{m+n}\)
• Quotient of Powers Rule: \(\frac{x^m}{x^n} = x^{m-n}\)
• Power of a Power Rule: \((x^m)^n = x^{m \cdot n}\)
• Negative Exponent Rule: \(x^{-n} = \frac{1}{x^n}\)
• Zero Exponent Rule: \(x^0 = 1\) (for any non-zero x)

By applying these rules, we can simplify and make complex expressions more comprehensible. For our exercise, the negative exponent rule was crucial in turning \(a^{-7}\) into \(\frac{1}{a^7}\). Understanding and applying these rules will make dealing with exponents far easier.

Simplifying expressions is a process that makes mathematical expressions more compact and easier to understand or solve. This often involves applying arithmetic operations, algebraic rules, and properties such as the exponent rules.

In our given exercise, after converting \(a^{-7}\) to \(\frac{1}{a^7}\), we replaced the term in the initial expression \(5 a^{-7} b^4\), yielding \(5 b^4 \cdot \frac{1}{a^7}\). Simplification means adjusting this to its most concise form, where we combine like terms or restructure the expression.

Therefore, the final simplified form for our example is \(\frac{5 b^4}{a^7}\). This transformation is vital as it turns a more complex, hard-to-read expression into something more straightforward and easier to work with. Simplifying expressions is a key skill for solving algebraic problems efficiently.