APPROACH #1: Number sense

First recognize that \(79^{43}\) = \(79^{31} \times 79^{12}\)

While \(79^{31}\) might be a little bit smaller than 80^{31}, once we multiply \(79^{31}\) by \(79^{12}\) (a very large number), the resulting \(79^{43}\) will definitely be greater than \(80^{31}\)

Answer: A


APPROACH #2: Exponent laws and inequality laws
Let's first start with some matching operations (see video below on this technique)

Given:
Quantity A: \(79^{43}\)
Quantity B: \(80^{31}\)

Divide both quantities by \(79^{31}\) to get:

Quantity A: \(\frac{79^{43}}{79^{31}}\)

Quantity B: \(\frac{80^{31}}{79^{31}}\)


Simplify to get:
Quantity A: \(79^{12}\)

Quantity B: \((\frac{80}{79})^{31}\)

At this point we're comparing \(79^{12}\) with \((\frac{80}{79})^{31}\)

First notice that \(64^{12} < 79^{12}\)

Now rewrite \(64\) and \(2^6\) to get: \((2^6)^{12} < 79^{12}\)

Simplify \((2^6)^{12}\) to get: \(2^{72} < 79^{12}\)

Since \(\frac{80}{79} < 2\), we know that \((\frac{80}{79})^{72} < 2^{72}\)

Finally, since \((\frac{80}{79})^{31} < (\frac{80}{79})^{72}\), we can write: \((\frac{80}{79})^{31} < (\frac{80}{79})^{72} < 2^{72} < 79^{12}\)

This tells us that: \((\frac{80}{79})^{31} < 79^{12}\)

Answer: A


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