Value Comparison Exercise: Identifying the Maximum Number

Choose the largest value:

We need to find the largest of the given roots of 64:

• Calculate $\sqrt[6]{64}$:
  $\sqrt[6]{64} = 64^{1/6}$
  Since $64 = 2^6$, we have:
  $(2^6)^{1/6} = 2$

• Calculate $\sqrt[4]{64}$:
  $\sqrt[4]{64} = 64^{1/4}$
  Using the exponent $64 = 2^6$, we get:
  $(2^6)^{1/4} = 2^{6/4} = 2^{1.5} = \sqrt[2]{8} = 2 \times \sqrt{2} \approx 2.828$

• Calculate $\sqrt[3]{64}$:
  $\sqrt[3]{64} = 64^{1/3}$
  This simplifies to:
  $(2^6)^{1/3} = 2^{6/3} = 2^2 = 4$

• Calculate $\sqrt[]{64}$:
  $\sqrt[]{64} = 64^{1/2}$
  This gives us:
  $(2^6)^{1/2} = 2^{6/2} = 2^3 = 8$

Now, let's compare these calculated values:
- $\sqrt[6]{64} = 2$
- $\sqrt[4]{64} \approx 2.828$
- $\sqrt[3]{64} = 4$
- $\sqrt[]{64} = 8$

Among these values, the largest value is $\sqrt[]{64}$, which equals 8.

Therefore, the largest value is $\sqrt[]{64}$.