Verified Answer :

  • Apply the power of a product rule to distribute the exponent \(-3\): \(4^{-3} \cdot (m^3)^{-3} \cdot (n^{-4})^{-3}\).
  • Use the power of a power rule to multiply the exponents: \(4^{-3} \cdot m^{-9} \cdot n^{12}\).
  • Convert negative exponents to positive ones: \(\frac{1}{4^3} \cdot \frac{1}{m^9} \cdot n^{12}\).
  • Simplify the numerical coefficient and combine the terms into a single fraction: \(\frac{n^{12}}{64 m^9}\).
  • The final simplified expression is \(\boxed{\frac{n^{12}}{64 m^9}}\).

Explanation

  1. Understanding the Rules of Exponents

To simplify the expression \(\left(4 m^3 n^{-4}\right)^{-3}\), we utilize fundamental laws of exponents. Our goal is to rewrite the expression such that all exponents are positive integers. The key properties of exponents we will employ are:

  1. The power of a product rule: \((abc)^k = a^k b^k c^k\).
  2. The power of a power rule: \((a^m)^n = a^{m \cdot n}\).
  3. The definition of negative exponents: \(a^{-k} = \frac{1}{a^k}\), which implies that terms with negative exponents can be moved to the denominator (or numerator if they were in the denominator) to make the exponent positive.
  1. Applying Exponent Rules

We first apply the power of a product rule, \((abc)^k = a^k b^k c^k\), to distribute the outer exponent of \(-3\) to every factor inside the parentheses:
\[ \left(4 m^3 n^{-4}\right)^{-3} = 4^{-3} \cdot (m^3)^{-3} \cdot (n^{-4})^{-3} \]
Next, we apply the power of a power rule, \((a^m)^n = a^{m \cdot n}\), to calculate the exponents for each variable:
\[ 4^{-3} \cdot m^{3 \cdot (-3)} \cdot n^{-4 \cdot (-3)} = 4^{-3} \cdot m^{-9} \cdot n^{12} \]
We now have the expression \(4^{-3} \cdot m^{-9} \cdot n^{12}\).

  1. Final Simplification

Now, we convert the negative exponents to positive exponents using the property \(a^{-k} = \frac{1}{a^k}\).
For the coefficient \(4^{-3}\):
\[ 4^{-3} = \frac{1}{4^3} = \frac{1}{64} \]
For the variable term \(m^{-9}\):
\[ m^{-9} = \frac{1}{m^9} \]
The term \(n^{12}\) already has a positive exponent, so it remains in the numerator. Putting it all together:
\[ 4^{-3} \cdot m^{-9} \cdot n^{12} = \frac{1}{64} \cdot \frac{1}{m^9} \cdot n^{12} = \frac{n^{12}}{64 m^9} \]
This result contains only positive exponents, which satisfies the problem's requirements.

Examples

Exponents are widely used in scientific notation and engineering to handle very large or very small numbers, such as in physics calculations involving distance or force. For instance, if a formula describes the scaling of a material's strength based on its dimensions, one might encounter an expression like \((2x^2 y^{-3})^{-2}\). By simplifying this using exponent rules, an engineer can more easily determine how changing a physical dimension affects the overall performance of the system, ensuring efficiency and accuracy in design.

Related Questions