Verified Answer :

  • Decompose the terms inside the square root to identify perfect squares: \(16=4^2\), \(s^5=s^4 \cdot s\), and \(t^7=t^6 \cdot t\).
  • Separate the perfect square factors from the remaining radical: \(\sqrt{4^2 \cdot (s^2)^2 \cdot (t^3)^2 \cdot s \cdot t}\).
  • Simplify the square roots of the perfect square components: \(4 \cdot s^2 \cdot t^3 \sqrt{st}\).
  • State the final result: \(\boxed{4s^2 t^3 \sqrt{st}}\).

Explanation

  1. Problem Analysis and Objective

The objective is to simplify the expression \(\sqrt{16 s^5 t^7}\), where the variables \(s\) and \(t\) are positive real numbers. We will use the properties of square roots and exponent laws to extract the perfect square factors from the expression inside the radical. Recall that \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) and that \(\sqrt{x^n} = x^{n/2}\) for \(x > 0\).

  1. Factoring into Perfect Squares

We look for the largest even powers of \(s\) and \(t\) that can be factored out. The expression is \(\sqrt{16 s^5 t^7}\).
We can rewrite the components:

  • \(16 = 4^2\)
  • \(s^5 = s^4 \cdot s = (s^2)^2 \cdot s\)
  • \(t^7 = t^6 \cdot t = (t^3)^2 \cdot t\)

So, the full expression under the square root is \(4^2 \cdot (s^2)^2 \cdot (t^3)^2 \cdot s \cdot t\).

  1. Simplifying the Radical

Now we apply the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) to the expression.
\[ \sqrt{4^2 \cdot (s^2)^2 \cdot (t^3)^2 \cdot s \cdot t} = \sqrt{4^2} \cdot \sqrt{(s^2)^2} \cdot \sqrt{(t^3)^2} \cdot \sqrt{s \cdot t} \]
Since \(s, t > 0\), the square roots of the squares simplify nicely:

  • \(\sqrt{4^2} = 4\)
  • \(\sqrt{(s^2)^2} = s^2\)
  • \(\sqrt{(t^3)^2} = t^3\)

Thus, the expression simplifies to \(4 \cdot s^2 \cdot t^3 \cdot \sqrt{st}\).

  1. Final Conclusion

Combining the factors obtained, we reach the simplified form of the original expression. The simplified radical is \(4s^2 t^3 \sqrt{st}\). Therefore, \(\sqrt{16 s^5 t^7} = \boxed{4s^2 t^3 \sqrt{st}}\).

Examples

This type of simplification is used frequently in engineering and physics when calculating formulas involving dimensions, such as area or volume. For instance, if you are calculating the volume of a geometric shape where the dimensions are expressed in algebraic terms, simplifying these radicals allows for cleaner, more readable equations that are easier to work with in further calculations or when plotting data on graphs. It helps in effectively managing units and scaling physical properties in real-world modeling.

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