Problem 5 - Algebra 1

Question

Use the quotient of powers property to simplify the expression. $$ \frac{a^{12}}{a^{9}} $$

Step-by-Step Solution

Answer

The simplified form of \(a^{12}/a^{9}\) is \(a^3\)

Step 1: Identify the Exponents

Here, in the expression \(a^{12}/a^{9}\), 'm' is 12 and 'n' is 9.

Step 2: Apply the Quotient of Powers Property

Using the property, subtract the exponent of the denominator from the exponent of the numerator. The new expression is \(a^{12-9}\) .

Step 3: Simplify the Expression

Subtract the exponents to simplify further. So, \(a^{12-9} = a^3\)

Key Concepts

Understanding Exponents

Exponents refer to the number of times a number, or base, is multiplied by itself. They appear in the form of a small number (the exponent) positioned to the top right of the base. For example, in the expression \(a^{12}\), 12 is the exponent, and it signifies that "a" is multiplied by itself 12 times:

• \(a^{12} = a \times a \times a \times \ldots \times a\) (12 times)

Exponents are a way to express repeated multiplication concisely. They play a crucial role in simplifying mathematical expressions. By dealing with them, we follow certain rules, such as what to do when multiplying, dividing, or raising powers to powers.

Simplifying Expressions with Quotient of Powers Property

Simplifying expressions can sometimes seem daunting, especially when dealing with large exponents. However, using the properties of exponents can make this task much easier. One such property is the Quotient of Powers Property.
This property states that when you divide like bases with exponents, you subtract the exponent of the denominator from the exponent of the numerator:

• \(\frac{a^m}{a^n} = a^{m-n}\)

Using this property transforms complex divisions into simpler forms. By applying it, the expression \(\frac{a^{12}}{a^9}\) becomes \(a^{12-9}\). This step reduces a seemingly complex expression to a much simpler one, making it easier to handle.

Subtraction of Exponents

Subtraction of exponents is a straightforward process once you recognize it as part of the Quotient of Powers Property. After identifying the bases as the same, you focus on the exponents in the numerator and the denominator.
In our previous example, we see an expression of \(\frac{a^{12}}{a^9}\). By applying subtraction of exponents, we perform:

• \(12 - 9 = 3\)

This operation shows clearly that the power of "a" simplifies from a high exponent to a significantly lower one, resulting in \(a^3\). Thus, subtracting exponents is an efficient method to break down exponential expressions into simpler components, making calculations manageable and less time-consuming.