1.7.6 stepping stones\nselect all the expressions equivalent to $(100)^{\frac{6}{7}}$.\n$\\left((100)^{\frac{…

1.7.6 stepping stones\nselect all the expressions equivalent to $(100)^{\frac{6}{7}}$.\n$\\left((100)^{\frac{1}{6}}\\right)^{7}$\n$\\sqrt7{(100)^{6}}$\n$\\left((100)^{7}\\right)^{\frac{1}{6}}$\n$\\left((100)^{6}\\right)^{\frac{1}{7}}$\n$\\left((100)^{\frac{1}{7}}\\right)^{6}$\n$\\sqrt6{(100)^{7}}$

1.7.6 stepping stones\nselect all the expressions equivalent to $(100)^{\frac{6}{7}}$.\n$\\left((100)^{\frac{1}{6}}\\right)^{7}$\n$\\sqrt7{(100)^{6}}$\n$\\left((100)^{7}\\right)^{\frac{1}{6}}$\n$\\left((100)^{6}\\right)^{\frac{1}{7}}$\n$\\left((100)^{\frac{1}{7}}\\right)^{6}$\n$\\sqrt6{(100)^{7}}$

Answer

Explanation:

Step1: Recall exponent-root rules

For any positive number $a$, $a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(a^{\frac{1}{n}}\right)^m = \left(a^m\right)^{\frac{1}{n}}$.

Step2: Match to target expression

Target: $(100)^{\frac{6}{7}}$

  • $\sqrt[7]{(100)^6}$: Matches $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ where $a=100, m=6, n=7$.
  • $\left((100)^6\right)^{\frac{1}{7}}$: Matches $a^{\frac{m}{n}} = \left(a^m\right)^{\frac{1}{n}}$.
  • $\left((100)^{\frac{1}{7}}\right)^6$: Matches $a^{\frac{m}{n}} = \left(a^{\frac{1}{n}}\right)^m$.

Step3: Eliminate non-equivalent expressions

  • $\left((100)^{\frac{1}{6}}\right)^7 = (100)^{\frac{7}{6}}$: Not equivalent.
  • $\left((100)^7\right)^{\frac{1}{6}} = (100)^{\frac{7}{6}}$: Not equivalent.
  • $\sqrt[6]{(100)^7} = (100)^{\frac{7}{6}}$: Not equivalent.

Answer:

$\sqrt{(100)^6}^7$, $\left((100)^6\right)^{\frac{1}{7}}$, $\left((100)^{\frac{1}{7}}\right)^6$