91Ó°ÊÓ

Simplifying expressions, particularly those involving exponents, involves rewriting them in a form that's easier to work with using known algebraic rules. For the given exercise, we start with the expression \(x^{-5} + \frac{1}{x^{8}}\). Our goal is to rewrite it in its simplest form without changing its value. By simplifying such expressions:

• We make calculations and further mathematical operations more straightforward.
• We help prevent errors in more complex equations.

To simplify, we apply exponent rules to each term of the expression separately, aiming for a format that's generally more manageable and intuitive.

Negative exponents might seem tricky at first, but they actually have a simple rule that makes them easier to handle. The rule states that any number with a negative exponent \(x^{-n}\) is equivalent to \(\frac{1}{x^n}\), where \(n\) is a positive integer. This transformation helps to convert a number with a negative exponent into a fraction, bringing it into a simpler and more familiar form.For the first term in our exercise, \(x^{-5}\) becomes \(\frac{1}{x^5}\) after applying this rule. The second term \(\frac{1}{x^8}\) already follows the format derived from this rule. Understanding negative exponents is crucial, as it frequently appears in algebraic expressions and equations, making it essential for effective problem-solving.

Exponents follow specific rules that allow us to manipulate and simplify expressions. Here are some core rules:

• Product of Powers Rule: Multiply exponents when the bases are the same \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers Rule: Subtract the exponents when dividing like bases \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power Rule: Multiply the exponents \((a^m)^n = a^{m \cdot n}\).
• Negative Exponent Rule: As aforementioned, it allows conversion to reciprocals \(a^{-n} = \frac{1}{a^n}\).
• Zero Exponent Rule: Any number raised to the zero power \(a^0 = 1\).

In our exercise, the negative exponent rule is utilized most, turning \(x^{-5}\) into \(\frac{1}{x^5}\). Familiarity with these rules is vital in simplifying expressions and solving equations effectively.