\(\sqrt[5]{70\mathrm{n}} \cdot (\sqrt[6]{70\mathrm{n}})^2\)For what value of x is the given expression equivalent to \((70\mathrm{n})^{30\mathrm{x}}\)...

1. TRANSLATE the radical expressions to exponential form

• Given information:
  • Expression: \(\sqrt[5]{70n}(\sqrt[6]{70n})^2\)
  • This must equal: \((70n)^{30x}\)

• TRANSLATE using the rule \(\sqrt[n]{a} = a^{1/n}\):
  • \(\sqrt[5]{70n} = (70n)^{1/5}\)
  • \(\sqrt[6]{70n} = (70n)^{1/6}\)
  • \((\sqrt[6]{70n})^2 = ((70n)^{1/6})^2 = (70n)^{2/6} = (70n)^{1/3}\)

2. SIMPLIFY the expression using exponent rules

• The expression becomes: \((70n)^{1/5} \cdot (70n)^{1/3}\)

• Using \(a^m \cdot a^n = a^{m+n}\):
  \((70n)^{1/5 + 1/3}\)

• Add the fractions: \(1/5 + 1/3 = 3/15 + 5/15 = 8/15\)

• Result: \((70n)^{8/15}\)

3. INFER the equation and solve

• Since \((70n)^{8/15} = (70n)^{30x}\), the exponents must be equal:
  \(8/15 = 30x\)

• SIMPLIFY to solve for x:
  \(x = 8/(15 \times 30) = 8/450 = 4/225\)

4. Verify the decimal form (use calculator)

Answer: 4/225, 0.017, 0.018, .0177, .0178

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly convert \((\sqrt[6]{70n})^2\) to \((70n)^{2/6}\) and forget to simplify \(2/6\) to \(1/3\), instead keeping it as \((70n)^{2/6}\) and getting the wrong final exponent.

This leads to setting up the equation as \((70n)^{1/5 + 2/6} = (70n)^{30x}\), where \(1/5 + 2/6 = 6/30 + 10/30 = 16/30 = 8/15\). Ironically, this still gives the correct answer, but represents flawed understanding of exponent simplification.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when adding fractions \(1/5 + 1/3\), often getting wrong results like \(2/8\) or \(1/2\), leading to incorrect final answers.

This causes them to get stuck and guess from the available answer formats.

The Bottom Line:

This problem tests your ability to work fluently between radical and exponential notation while maintaining accuracy with fraction arithmetic. The key insight is recognizing that radical expressions are just another way to write fractional exponents, making the algebra much more manageable.