Value Comparison Exercise: Identifying the Maximum Number

Question

Choose the largest value:

Video Solution

Solution Steps

00:00 Choose the largest value
00:03 A 6th root is like a power with the reciprocal of 6 (one-sixth)
00:07 Apply the same method to all expressions and determine the largest value
00:23 A "regular" root raised to the second power
00:28 This is the solution

Step-by-Step Solution

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

  • Calculate 646\sqrt[6]{64}:
    646=641/6 \sqrt[6]{64} = 64^{1/6}
    Since 64=2664 = 2^6, we have:
    (26)1/6=2 (2^6)^{1/6} = 2

  • Calculate 644\sqrt[4]{64}:
    644=641/4 \sqrt[4]{64} = 64^{1/4}
    Using the exponent 64=2664 = 2^6, we get:
    (26)1/4=26/4=21.5=82=2×22.828 (2^6)^{1/4} = 2^{6/4} = 2^{1.5} = \sqrt[2]{8} = 2 \times \sqrt{2} \approx 2.828

  • Calculate 643\sqrt[3]{64}:
    643=641/3 \sqrt[3]{64} = 64^{1/3}
    This simplifies to:
    (26)1/3=26/3=22=4 (2^6)^{1/3} = 2^{6/3} = 2^2 = 4

  • Calculate 64\sqrt[]{64}:
    64=641/2 \sqrt[]{64} = 64^{1/2}
    This gives us:
    (26)1/2=26/2=23=8 (2^6)^{1/2} = 2^{6/2} = 2^3 = 8

Now, let's compare these calculated values:
- 646=2\sqrt[6]{64} = 2
- 6442.828\sqrt[4]{64} \approx 2.828
- 643=4\sqrt[3]{64} = 4
- 64=8\sqrt[]{64} = 8

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

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

Answer

64 \sqrt[]{64}