Question

Use the quotient of powers property to simplify the expression. $$ \frac{a^{5}}{a^{2}} $$

Short answer

The simplified form of the expression \(\frac{a^{5}}{a^{2}}\) is \(a^{3}\)

Step-by-step solution

Recognize the Quotient of Powers Property

We have an expression of the form \(a^n/a^m\), which represents the quotient of powers property. This property states that for any number a and integers m and n, \(a^{m}/a^{n} = a^{m-n}\)

Apply the Quotient of Powers Property

In our expression \(\frac{a^{5}}{a^{2}}\), simply subtract the exponent in the denominator from the one in the numerator, following the quotient of powers rule: so \(a^{5-2} = a^{3}\)

Key concepts

Exponents

Exponents are a compact way to represent repeated multiplication of the same number. For example, the expression \(a^{5}\) signifies multiplying the base \(a\) by itself four more times, i.e., \(a \times a \times a \times a \times a\). Understanding how exponents work is crucial when simplifying expressions involving power operations.

When dealing with quotients of a similar base, the Quotient of Powers Property allows us to subtract exponents: \(\frac{a^{m}}{a^{n}} = a^{m-n}\), provided that \(a\) is not zero. This rule simplifies calculations and helps maintain expression integrity without expanding the terms which can be lengthy and more complicated. Students should comfort themselves with the fundamentals of working with exponents, including the product of powers, the power of a power, and the power of a product to tackle algebra problems efficiently.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill in mathematics, as it allows for clearer and more concise representations of problems. The process involves reducing an expression to its simplest form by performing operations such as adding like terms, factoring, multiplying polynomials, and using algebraic properties.

In the context of exponents, simplification typically involves applying rules like the aforementioned Quotient of Powers. This simplification can make algebraic expressions more straightforward, thus making them more approachable for further analysis or problem-solving. Students are encouraged to practice these simplifications to become proficient in identifying opportunities to reduce expressions and execute the necessary steps for simplification confidently.

Algebraic Properties

Algebraic properties are the foundation that allows us to manipulate and solve equations and expressions. There are several key properties, such as the Commutative Property (allowing the rearrangement of terms), Associative Property (allowing the regrouping of terms), and Distributive Property (combining multiplication across addition or subtraction).

The Quotient of Powers Property falls under these fundamental guidelines, highlighting the predictable behavior of exponents during division. When students grasp how algebraic properties simplify and reorganize mathematical expressions, they gain powerful tools to handle more complex algebraic challenges. Recognizing and correctly applying these properties are the building blocks for understanding higher-level math and are invaluable in a range of academic and real-world applications.