Problem 42 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \frac{15 a^{4}}{-15 a^{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-1}{a} \).

Step 1: Divide the coefficients

The expression \( \frac{15 a^4}{-15 a^5} \) has coefficients 15 in the numerator and -15 in the denominator. Divide the coefficients: \( \frac{15}{-15} = -1 \). This simplifies the expression to \( -1 \cdot \frac{a^4}{a^5} \).

Step 2: Apply the properties of exponents

Use the property \( \frac{a^m}{a^n} = a^{m-n} \) to simplify the variable part. Here, \( m = 4 \) and \( n = 5 \), so \( \frac{a^4}{a^5} = a^{4-5} = a^{-1} \). Now, our expression is \( -1 \cdot a^{-1} \).

Step 3: Convert to positive exponents

Rewrite \( a^{-1} \) using a positive exponent: \( a^{-1} = \frac{1}{a} \). The expression is now \( -1 \cdot \frac{1}{a} = \frac{-1}{a} \).

Key Concepts

Properties of Exponents

When dealing with exponents, several properties help us simplify expressions effectively. One fundamental property is the **quotient of powers rule:** if you have an expression of the form \( \frac{a^m}{a^n} \), where \( a \) is a non-zero base and \( m \) and \( n \) are integers, you can simplify it to \( a^{m-n} \).
This property makes it easier to manage expressions by reducing the exponent's size, as it combines the effect of division into a single exponentiation. For example, in the expression \( \frac{a^4}{a^5} \), applying this rule gives us \( a^{4-5} = a^{-1} \).

• Remember that this rule applies only when the two bases are the same.
• Make sure you subtract the exponent in the denominator from the exponent in the numerator.

Understanding these properties helps in simplifying both simple and complex algebraic expressions.

Positive Exponents

Exponents can be either positive or negative, and it's often useful to express results using only positive exponents. A negative exponent, such as \( a^{-1} \), indicates a reciprocal. Therefore, \( a^{-1} \) means \( \frac{1}{a} \).
This approach helps when you need to ensure standard expression forms or apply further algebraic simplifications more systematically. In the simplified expression \( -1 \cdot a^{-1} \), turning \( a^{-1} \) into \( \frac{1}{a} \) results in \( \frac{-1}{a} \).

• Positive exponents signify normal multiplication, while negative exponents signify taking reciprocals.
• To convert a negative exponent to positive, simply take the reciprocal of the base raised to the positive exponent.

Maintaining positive exponents provides cleaner results in the context of real-world applications and theoretical mathematics.

Division of Algebraic Expressions

Dividing algebraic expressions involves working with both coefficients and variables that have exponents. You start by separately dividing the numerical coefficients and applying the properties of exponents to the variables. Let's revisit our expression \( \frac{15 a^4}{-15 a^5} \).
First, divide the coefficients: \( \frac{15}{-15} = -1 \).
Next, use the properties of exponents to divide the variables: \( \frac{a^4}{a^5} = a^{4-5} = a^{-1} \). Combining these, the expression simplifies to \( -1 \cdot a^{-1} \), which we rewrite using positive exponents as \( \frac{-1}{a} \).

• Always divide coefficients directly, treating them like regular numbers.
• Apply the quotient rule for exponents to simplify variable parts.
• Rewrite the result to include only positive exponents for final simplification.

Mastering division of algebraic expressions streamlines the simplification process and enhances overall problem-solving efficiency.