Question

Simplify each expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not \(0 .\) \(\frac{x^{2} y^{5}}{x^{3} y^{4}}\)

Short answer

\[ \frac{y}{x} \]

Step-by-step solution

Write down the given expression

Start with the given expression: \[ \frac{x^{2} y^{5}}{x^{3} y^{4}} \]

Apply the properties of exponents

Use the rule \( \frac{a^{m}}{a^{n}} = a^{m-n} \) to simplify both the numerator and the denominator. For the x terms: \[ x^{2-3} = x^{-1} \], and for the y terms: \[ y^{5-4} = y^{1} \]

Simplify and ensure positive exponents

Rewrite \( x^{-1} \) using positive exponents: \[ x^{-1} = \frac{1}{x} \]. Therefore, the expression becomes: \[ \frac{y^{1}}{x} = \frac{y}{x} \]

Key concepts

Properties of Exponents

When simplifying expressions with exponents, understanding their properties is essential. One critical property is the quotient of powers property, which states: \[ \frac{a^m}{a^n} = a^{m-n} \] This property helps us simplify expressions where the same base appears both in the numerator and the denominator. For instance, in the expression \[ \frac{x^2 y^5}{x^3 y^4} \] we can apply this property to both the x and y terms. Applying the property to the x terms gives \[ x^{2-3} = x^{-1} \] and to the y terms, we get \[ y^{5-4} = y \] It's crucial to apply these properties correctly to simplify the expression without errors.

Positive Exponents

To make expressions easier to work with, we often convert all exponents to positive values. Negative exponents can be rewritten using positive exponents by the rule: \[ a^{-n} = \frac{1}{a^n} \] For example, in the expression \[ x^{-1} \] it can be rewritten as \[ \frac{1}{x} \] This is a key step in our solution to ensure all exponents are positive. Changing negative exponents to positive aids in simplifying and interpreting the expressions correctly.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their most basic form. This includes combining like terms and applying the properties of exponents, as seen in our exercise. Start with the given expression: \[ \frac{x^2 y^5}{x^3 y^4} \] Apply the quotient of powers property: \[ x^{2-3} = x^{-1} \] and \[ y^{5-4} = y \] Then, convert any negative exponents to positive: \[ x^{-1} = \frac{1}{x} \] So the simplified expression becomes: \[ \frac{y}{x} \] By breaking down the steps and adhering to these rules, simplifying algebraic expressions becomes more manageable. Practice is key to mastering this skill.