Answered

Simplify the following expression to simplest form using only positive exponents.



$\left(4 x^{14} y^{-4}\right)^{\frac{3}{2}}$

Verified Answer :

  • Apply the power of a product rule to the expression: \(4^{\frac{3}{2}} \cdot (x^{14})^{\frac{3}{2}} \cdot (y^{-4})^{\frac{3}{2}}\).
  • Simplify the constant term: \(4^{\frac{3}{2}} = 8\).
  • Simplify the variable exponents using the power of a power rule: \(x^{14 \cdot \frac{3}{2}} = x^{21}\) and \(y^{-4 \cdot \frac{3}{2}} = y^{-6}\).
  • Express the result with only positive exponents: \(\frac{8x^{21}}{y^6}\).
  • The final answer is \(\boxed{\frac{8x^{21}}{y^6}}\).

Explanation

  1. Analyzing the Expression

We are given the expression \(\left(4 x^{14} y^{-4}\right)^{\frac{3}{2}}\) and we need to simplify it into its simplest form using only positive exponents. The problem involves applying rules of exponents, specifically the power of a product rule, the power of a power rule, and the definition of negative exponents. We will break down the expression term by term.

  1. Simplifying Individual Terms

The power of a product rule states that \((abc)^n = a^n \cdot b^n \cdot c^n\). Applying this to our expression, we get:
\[\left(4 x^{14} y^{-4}\right)^{\frac{3}{2}} = 4^{\frac{3}{2}} \cdot (x^{14})^{\frac{3}{2}} \cdot (y^{-4})^{\frac{3}{2}}\]
Now, we simplify each factor individually:

  1. For the constant term: \(4^{\frac{3}{2}} = (\sqrt{4})^3 = 2^3 = 8\).
  2. For the \(x\) term: \(x^{14 \cdot \frac{3}{2}} = x^{21}\).
  3. For the \(y\) term: \(y^{-4 \cdot \frac{3}{2}} = y^{-6}\).

Combining these, we have \(8 \cdot x^{21} \cdot y^{-6}\).

  1. Final Simplification

Finally, we write the expression with only positive exponents. The term \(y^{-6}\) is equivalent to \(\frac{1}{y^6}\). Therefore, the simplified expression is:
\[8 x^{21} \cdot \frac{1}{y^6} = \frac{8 x^{21}}{y^6}\]
This is the simplest form using only positive exponents.

Examples

In fields like engineering and physics, formulas often involve quantities raised to fractional powers. For example, when calculating the volume or energy of a system where dimensions are scaled by a fractional factor, simplifying expressions allows for easier manipulation and clearer interpretation of the relationships between variables. This technique ensures that complex scientific equations can be reduced to manageable forms for calculation and analysis.

Related Questions