Problem 67 - Elementary Algebra

Question

The link between positive and negative exponents \(\left(a^{-n}=\frac{1}{a^{n}}\right)\) along with the property \(\frac{a^{n}}{a^{m}}=a^{n-m}\) can also be used when reducing fractions. Consider this example: $$ \frac{x^{3}}{x^{7}}=x^{3-7}=x^{-4}=\frac{1}{x^{4}} $$ Use this approach to express each fraction in reduced form. Give all answers with positive exponents only. $$\frac{x^{3}}{x^{9}}$$

Step-by-Step Solution

Answer

\( \frac{x^3}{x^9} = \frac{1}{x^6} \)

Step 1: Apply the Property of Exponents

According to the property \( \frac{a^n}{a^m} = a^{n-m} \), start by applying this rule to the given fraction \( \frac{x^3}{x^9} \). This simplifies to \( x^{3-9} = x^{-6} \).

Step 2: Convert Negative Exponent to Positive

Next, use the rule that \( a^{-n} = \frac{1}{a^n} \) to express \( x^{-6} \) with a positive exponent. This results in \( \frac{1}{x^6} \).

Key Concepts

Positive Exponents

Exponents are a way to express the repeated multiplication of a number by itself. Positive exponents indicate how many times to multiply the base number. For example, when we have \( x^3 \), it means \( x \cdot x \cdot x \), which is "x" multiplied by itself three times.
Positive exponents are quite straightforward because they show direct multiplication without any need for inversion. Using positive exponents helps to represent large numbers in a compact format.
It's essential to remember:

• In an expression such as \( a^n \), both \( a \) and \( n \) should be positive for clarifying multiplication without any inversions.
• Multiplying terms with the same base involves adding the exponents, e.g., \( x^2 \cdot x^3 = x^{2+3} = x^5 \).

Negative Exponents

Negative exponents might seem tricky at first, but they simplify expressions by indicating division instead of multiplication. The rule to remember is that a negative exponent equals the reciprocal with a positive exponent. For example, \( a^{-n} = \frac{1}{a^n} \).
Consider \( x^{-4} = \frac{1}{x^4} \). Here, the negative sign flips the base onto the opposite side of a fraction, making any expression with negative exponents easier to interpret once you apply this rule.
Keep these points in mind:

• Negative exponents are about inversion. They don't mean the number itself is negative.
• Always convert negative exponents into positive exponents for simpler and more intuitive expressions.

Fraction Reduction

Reducing fractions in the context of exponents involves simplifying the fraction by applying the properties of exponents. A fraction like \( \frac{x^3}{x^9} \) can be reduced using exponent rules, making it easier to handle.
When you have two powers of the same base in a fraction, subtract the exponents: \( \frac{a^n}{a^m} = a^{n-m} \). In our example, \( \frac{x^3}{x^9} = x^{3-9} = x^{-6} \).
Here’s how to approach it step-by-step:

• Identify the powers on the top and bottom of the fraction.
• Subtract the bottom exponent from the top exponent.
• Simplify to see if a positive exponent is possible, or convert negative exponents to positive.

Properties of Exponents

The properties of exponents simplify working with repeated multiplication and division of the same base. Key properties to remember include:
1. **Product of Powers:** \( a^n \cdot a^m = a^{n+m} \). Multiply two powers of the same base by adding the exponents.
2. **Quotient of Powers:** \( \frac{a^n}{a^m} = a^{n-m} \). Divide two powers of the same base by subtracting the exponents.
3. **Power of a Power:** \( (a^n)^m = a^{n \cdot m} \). When a power is raised to another power, multiply the exponents.
4. **Negative Exponents:** \( a^{-n} = \frac{1}{a^n} \). This property allows converting a negative exponent into a fraction.
5. **Zero Exponent:** Any base raised to the zero power is 1, e.g., \( a^0 = 1 \).
These properties make it easier to simplify expressions, especially in algebra, ensuring your calculations are both concise and correct.