Explanation

You can simplify both quantities. Quantity A can be simplified as follows:

\(80^1^/^3 = 8^1^/^3 * 10^1^/^3 = 2 *10^1^/^3\)

Quantity B can be simplified as follows:

\(270^1^/^3 - 10^1^/^3 = 27^1^/^3 *10^1^/^3 - 10^1^/^3= 3*10^1^/^3 - 10^1^/^3= 2 *10^1^/^3\)

Hence both Quantities are equal.
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IMPORTANT RULE:(ab)^y = (a^y)(b^y)
For example, 12^17 = (4^17)(3^17)

Given:
Quantity A: 80^(1/3)
Quantity B: 270^(1/3) - 10^(1/3)

Apply law to get:
Quantity A: [10^(1/3)][8^(1/3)]
Quantity B: [27^(1/3)][10^(1/3)] - 10^(1/3)

Factor out 10^(1/3) in quantity B to get:
Quantity A: [10^(1/3)][8^(1/3)]
Quantity B: [10^(1/3)][27^(1/3) - 1]

Divide both quantities by 10^(1/3) to get:
Quantity A: 8^(1/3)
Quantity B: 27^(1/3) - 1

Evaluate both quantities to get:
Quantity A: 2
Quantity B: 3 - 1

Answer:
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In one of the steps in Brent's solution, I didn't understand how the below Quantity B came to be. Can that be explained to me? Thanks.

Factor out 10^(1/3) in quantity B to get:
Quantity A: [10^(1/3)][8^(1/3)]
Quantity B: [10^(1/3)][27^(1/3) - 1]

In the step before that, we have: Quantity B: [27^(1/3)][10^(1/3)] - 10^(1/3)

This similar to the expression xy - y
With this expression, we can factor out the y to get: xy - y = y(x - 1)

Likewise, in the original expression, we can factor out 10^(1/3) to get: [27^(1/3)][10^(1/3)] - 10^(1/3) = [10^(1/3)][27^(1/3) - 1]

Does that help?

Cheers,
Brent
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