Question

Can \(\frac{x^{8}}{y^{3}}\) be simplified? Explain your answer.

Short answer

No, \(\frac{x^{8}}{y^{3}}\) cannot be simplified as there are no common factors between the numerator and the denominator. The expression is already in its simplest form.

Step-by-step solution

Identify the terms in the expression

Here, the numerator is \(x^{8}\) and the denominator is \(y^{3}\). These are the only terms in the expression.

Determine if the terms have common factors

\(x^{8}\) is a power of \(x\) and \(y^{3}\) is a power of \(y\). They do not share any common factors because \(x\) and \(y\) are different variables.

Determine if the expression can be simplified

True mathematical simplification involves cancelling out common factors between the numerator and the denominator. Since the terms in the expression do not share any common factors, \(\frac{x^{8}}{y^{3}}\) cannot be simplified. The expression is already in its simplest form.

Key concepts

Powers and Exponents

Understanding powers and exponents is essential for simplifying algebraic expressions. An exponent represents how many times a base number is multiplied by itself. For instance, in the expression x^{8}, the number 8 is an exponent, indicating that x, the base, is multiplied by itself eight times. It's crucial to grasp that each variable and its exponent are treated as unique terms in algebra.

When dealing with powers and exponents during simplification, you can only combine like terms. Like terms have the same base and exponent. For example, x^{2} can be combined with 3x^{2} but not with x^{3} or y^{2}. In our original exercise, x^{8} and y^{3} have different bases and therefore cannot be simplified by combining like terms.

Furthermore, the Laws of Exponents are pivotal in simplifying expressions involving powers. Among these laws are:

Variables in Algebra

Variables are symbols, typically letters like x and y, that represent numbers in algebraic expressions. When simplifying expressions, it's important to understand that each variable is unique and represents a potentially different value unless otherwise stated. Variables allow for a general form of expression that can be applied to many different situations.

In the case of simplifying the expression \(\frac{x^{8}}{y^{3}}\), x and y are distinct variables. Since they stand for different, unspecified values, they cannot be combined or simplified with respect to each other. This distinction is what keeps the expression in the simplest form as is. Simplification can occur when the same variable is present in both the numerator and denominator or through the use of algebraic operations if applicable.

Simplification of Expressions

The simplification of algebraic expressions involves reducing them to their simplest form without changing their value. This process might include combining like terms, factoring, expanding expressions, and canceling common factors. A common factor is a term that is present in both the numerator and the denominator that can be divided out.

In our exercise, since there are no common factors, as x and y are not the same, the expression remains as is. Simplification may seem like it always leads to a shorter expression, but this is not necessarily the case. The goal is to make the expression as clear and as straightforward as possible, which in our example, means acknowledging that \(\frac{x^{8}}{y^{3}}\) is already in its most simplified form because no further simplification rules apply.