Math Problem Statement

Simplification with Negative Indices

Solution

When simplifying expressions with negative indices (exponents), the key is to remember that a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. Here are the fundamental rules and steps to simplify such expressions:

Key Rules for Negative Indices

  1. Reciprocal Rule: an=1ana^{-n} = \frac{1}{a^n} This rule states that a term with a negative exponent can be rewritten as the reciprocal (or inverse) of the term with a positive exponent.

  2. Multiplication with Same Base: am×an=am+na^m \times a^n = a^{m+n} When multiplying like bases, you add the exponents.

  3. Division with Same Base: aman=amn\frac{a^m}{a^n} = a^{m-n} When dividing like bases, you subtract the exponents.

  4. Power of a Power: (am)n=am×n(a^m)^n = a^{m \times n} When raising a power to another power, you multiply the exponents.

Example: Simplifying an Expression with Negative Indices

Let's consider an example:

23344261\frac{2^{-3} \cdot 3^4}{4^{-2} \cdot 6^{-1}}

Step 1: Simplify Negative Indices

First, rewrite the terms with negative exponents as reciprocals:

1233414216=342314216\frac{\frac{1}{2^3} \cdot 3^4}{\frac{1}{4^2} \cdot \frac{1}{6}} = \frac{\frac{3^4}{2^3}}{\frac{1}{4^2} \cdot \frac{1}{6}}

Step 2: Simplify the Denominator

Multiply the terms in the denominator:

3423×4261=3442623\frac{3^4}{2^3} \times \frac{4^2 \cdot 6}{1} = \frac{3^4 \cdot 4^2 \cdot 6}{2^3}

Step 3: Simplify Using Exponent Rules

Now, let's simplify by expanding or combining like bases if possible:

  • 42=(22)2=244^2 = (2^2)^2 = 2^4
  • Combine powers of 2 in the expression:

3424623=342436=34216\frac{3^4 \cdot 2^4 \cdot 6}{2^3} = 3^4 \cdot 2^{4-3} \cdot 6 = 3^4 \cdot 2^1 \cdot 6

Step 4: Final Simplification

Combine the constants:

3426=3412=8112=9723^4 \cdot 2 \cdot 6 = 3^4 \cdot 12 = 81 \cdot 12 = 972

So, the expression simplifies to:

972972

Conclusion

Negative indices can make an expression seem more complex, but by applying these rules, you can simplify it systematically. Always convert negative exponents into their positive equivalents by taking the reciprocal, and then use the laws of exponents to combine terms.

Do you have any other questions or need further clarification? Here are some questions related to this topic:

  1. What is the impact of negative exponents on fractional bases?
  2. How do you simplify expressions with both negative and positive exponents?
  3. Can you convert all expressions with negative exponents to have positive exponents only?
  4. What is the difference between simplifying and solving expressions with exponents?
  5. How do negative exponents affect the graph of exponential functions?

Tip: Always check whether you can combine like terms or reduce the expression further after simplifying negative indices, as this often leads to the simplest form.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Exponents
Negative Exponents
Reciprocal Rule
Exponent Rules

Formulas

-

Theorems

-

Suitable Grade Level

Grades 9-12