Math Problem Statement

( 1 2 ) 3𝑛 − ( 4 5 ) 3

Solution

It looks like you're dealing with an expression involving parentheses and exponents. The expression can be rewritten as:

(12)3n(45)3\left( \frac{1}{2} \right)^{3n} - \left( \frac{4}{5} \right)^3

Let's break this down:

  1. (12)3n\left( \frac{1}{2} \right)^{3n} is an expression where 12\frac{1}{2} is raised to the power of 3n3n.
  2. (45)3\left( \frac{4}{5} \right)^3 represents 45\frac{4}{5} raised to the power of 3.

Simplifying:

(45)3=4353=64125\left( \frac{4}{5} \right)^3 = \frac{4^3}{5^3} = \frac{64}{125}

Thus, the full expression becomes:

(12)3n64125\left( \frac{1}{2} \right)^{3n} - \frac{64}{125}

This is the simplified form unless more context (e.g., specific values for nn) is provided.

Would you like further details on how to solve this for specific values of nn or simplify further?


5 related questions:

  1. How do you handle fractional exponents in equations?
  2. What happens if n=1n = 1 in this expression?
  3. How do you approach expressions with multiple exponents?
  4. Can this expression be simplified further for specific values of nn?
  5. What is the significance of negative exponents in expressions?

Tip: When dealing with exponents, always remember to apply the power rule for fractions carefully, raising both numerator and denominator to the given power.

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Math Problem Analysis

Mathematical Concepts

Exponents
Fractions
Algebra

Formulas

a^m * a^n = a^(m+n)
(a/b)^n = a^n / b^n

Theorems

Laws of Exponents

Suitable Grade Level

Grades 8-10