Question

Write the expression as an equivalent expression in the form \(x^{n}\) and give the value for \(n\). $$ \frac{1}{x^{5}} $$

Short answer

Question: Rewrite the expression \(\frac{1}{x^{5}}\) in the form \(x^{n}\) and find the value of \(n\). Answer: The expression can be rewritten as \(x^{-5}\), and the value of \(n\) is -5.

Step-by-step solution

Write down the given expression

First, we need to note down the given expression: $$ \frac{1}{x^{5}} $$

Apply the negative exponent rule

Now, we will apply the negative exponent rule, which states that for any non-zero number \(a\) and an integer \(n\), we have: $$ \frac{1}{a^n} = a^{-n} $$ Using this rule, we can rewrite the given expression as: $$ x^{-5} $$

Identify the value of n

Now, we can clearly see that the equivalent expression is in the form of \(x^{n}\). We just have to identify the value of \(n\). In this case, \(n\) is equal to -5. So, the rewritten expression is: $$ x^{-5} $$ with \(n = -5\).

Key concepts

Exponent Rules

Exponent rules are foundational principles in mathematics that help us manipulate expressions involving powers. These rules include products of powers, power of a power, power of a product, and perhaps most intriguingly, the rule for negative exponents.

A negative exponent indicates that the base number should be moved to the denominator of a fraction and the exponent then made positive. For example, if we have a base number raised to a negative power like this: \(a^{-n}\), it can be written as \(\frac{1}{a^n}\). Conversely, if the expression is \(\frac{1}{a^n}\), it can be rewritten using a negative exponent as \(a^{-n}\).
This rule helps simplify expressions and solve equations more efficiently.

• An understanding of negative exponents allows for seamless conversion between forms.
• It is essential in simplifying complex rational expressions.

Grasping these rules is vital as they frequently appear in algebra and calculus.

Equivalent Expressions

When we talk about equivalent expressions, it involves transforming an expression into another form without changing its overall value. This is particularly useful in algebra and calculus, as it allows us to simplify, compare, or manipulate expressions effectively.

For example, the given expression \(\frac{1}{x^{5}}\) was transformed into an equivalent expression \(x^{-5}\) through the use of the negative exponent rule. Although they look different, these two expressions hold the same mathematical value.
Understanding equivalent expressions is crucial for:

• Simplifying complex algebraic expressions.
• Making calculations easier and presentations neater.
• Solving equations and inequalities efficiently.

These transformations are key skills in solving algebraic equations and often used to prove mathematical identities.

Simplifying Expressions

Simplifying expressions involves reducing the complexity of an expression while maintaining its original value. This often includes applying the exponent rules and transforming the expression into an equivalent form.

The process of converting \(\frac{1}{x^{5}}\) into \(x^{-5}\) is a prime example of simplification. By applying the negative exponent rule, the expression becomes not only easier to read, but also more suitable for further mathematical operations.
When simplifying, keep these strategies in mind:

• Look for common factors and try to combine like terms.
• Apply exponent rules to reduce powers of expressions.
• Transform fractions into exponential form where applicable.

Effective simplification makes solving difficult problems more manageable and ensures clearer communication of mathematical ideas.