Question

Simplify the expression. The simplified expression should have no negative exponents. $$\frac{x^{4}}{x^{5}}$$

Short answer

\(\frac{1}{x}\)

Step-by-step solution

Identify the base and the exponents

The expression is given as \( \frac{x^{4}}{x^{5}} \), where the base is \(x\), exponent of the numerator is 4 and that of the denominator is 5.

Apply the quotient rule of exponents

The quotient rule of exponents states that to divide expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator. That gives \( x^{4-5} = x^{-1}\).

Rewrite to eliminate the negative exponent

If a term with an exponent is on the wrong side of the fraction line, you can flip it to the other side by changing the sign of the exponent. So, we rewrite it as \( \frac{1}{x^{1}} \).

Key concepts

Quotient Rule of Exponents

Understanding the quotient rule of exponents is critical in simplifying algebraic expressions with exponents effectively. When you have an expression of the form \( \frac{a^m}{a^n} \), where \( a \) is any non-zero number and \( m \) and \( n \) are any integers, the quotient rule tells us that we can subtract the exponent of the denominator from the exponent of the numerator. The formula looks like this: \[ a^m / a^n = a^{m-n} \].

This rule applies only when the bases are the same, in this case, both being \( a \) in our example. To see this in action, let’s apply it to a simple expression such as \( \frac{x^{4}}{x^{5}} \). Following the quotient rule, we subtract 5 from 4, resulting in \( x^{4-5} = x^{-1} \). This subtraction simplifies the initial problem but also introduces a negative exponent, which leads us to our next crucial concept, dealing with negative exponents.

Negative Exponents

Dealing with negative exponents might initially seem challenging, but the concept is relatively straightforward. When you come across an expression like \( x^{-1} \), the negative exponent indicates that the base (in this case, \( x \)) should be on the opposite side of a fraction. Therefore, \( x^{-1} \) is the same as \( \frac{1}{x^{1}} \), or simply \( \frac{1}{x} \).

The general rule for negative exponents \( a^{-n} \) is that you can rewrite them as \( \frac{1}{a^{n}} \) if the base was in the numerator, or as \( a^{n} \) if the base was in the denominator. It’s easy to remember that a negative exponent just means you need to flip the base to the other side of a fraction and make the exponent positive. Once done, you've successfully removed the negative exponent and made the expression much easier to work with.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, operators, and variables. Expressions are the sentences of the algebraic language, and understanding how to work with them is fundamental to mastering algebra. Simplifying expressions, as we've been doing, often involves applying various rules of algebra, including the quotient rule for exponents or the rule for negative exponents.

When simplifying an expression like \( \frac{x^{4}}{x^{5}} \) it's helpful to recognize what each part of the expression represents. The 'x' is the variable, and it's our base. The numbers '4' and '5' are the exponents that tell us how many times to multiply the base by itself. As we perform simplifications, we must ensure that every step abides by the foundational rules of algebra. By doing so, an initially complex expression becomes far more readable and manageable, and this simplicity is the goal of algebra: to express quantities in the most understandable form possible.