Answered

Match each radical expression with its simplified answer:


  1. $\sqrt{m n^{15}}$
  2. $\sqrt{m^{12}}$
  3. $\sqrt{m^n}$
  4. $\sqrt{m^9}$

Verified Answer :

  • \(\sqrt{m n^{15}} = n^7 \sqrt{m n}\)
  • \(\sqrt{m^{12}} = m^6\)
  • \(\sqrt{m^n} = m^{n/2}\)
  • \(\sqrt{m^9} = m^4 \sqrt{m}\)

Explanation

  1. Analysis of Radical Simplification Rules

To simplify radical expressions of the form \(\sqrt{x^k}\), where \(x \ge 0\), we utilize the property \(\sqrt{x^k} = x^{k/2}\). If \(k\) is an even integer, this simplifies perfectly to \(x^{k/2}\). If \(k\) is an odd integer, we can decompose the expression: \(\sqrt{x^k} = \sqrt{x^{k-1} \cdot x} = x^{(k-1)/2} \sqrt{x}\). We will apply these rules to simplify each of the given expressions.

  1. Simplification of the Radical Expressions
  2. For \(\sqrt{m n^{15}}\): We can split this as \(\sqrt{m \cdot n^{14} \cdot n}\). Applying the square root to the perfect square part \(n^{14}\), we get \(n^7\sqrt{m n}\).
  3. For \(\sqrt{m^{12}}\): Since the exponent is even, we simply divide the exponent by 2: \(m^{12/2} = m^6\).
  4. For \(\sqrt{m^n}\): By the property of rational exponents, the square root of a power is the power divided by 2. Thus, \(\sqrt{m^n} = m^{n/2}\).
  5. For \(\sqrt{m^9}\): We can split this as \(\sqrt{m^8 \cdot m}\). Applying the square root to the perfect square part \(m^8\), we get \(m^4\sqrt{m}\).
  1. Final Results and Matching

The simplified forms of the expressions are:

  • \(\sqrt{m n^{15}} = n^7 \sqrt{m n}\)
  • \(\sqrt{m^{12}} = m^6\)
  • \(\sqrt{m^n} = m^{n/2}\)
  • \(\sqrt{m^9} = m^4 \sqrt{m}\)

Thus, the expressions are matched as follows:
\(\sqrt{m n^{15}} \to n^7 \sqrt{m n}\)
\(\sqrt{m^{12}} \to m^6\)
\(\sqrt{m^n} \to m^{n/2}\)
\(\sqrt{m^9} \to m^4 \sqrt{m}\)

Examples

Imagine you are an engineer designing a circuit where electrical resistance scales with certain parameters. You encounter a formula like \(\sqrt{m^9}\) while calculating energy dissipation. Simplifying it to \(m^4 \sqrt{m}\) allows you to quickly estimate the effect of changing the parameter \(m\) on your system, rather than calculating a complex radical every time. This technique is essential in physics and engineering for reducing complex formulas into manageable forms.

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