Exponential Ratios and Rational Numbers - PSAT Math

From the equation in the problem statement

Now squaring both sides we get

this is a quadratic equation which equals

and the factors of this equation are

This gives us .

However, if we plug these solutions back into the original equation, does not create an equality. Therefore, is an extraneous solution.

The conjugate of is .

Now multiply both the numerator and the denominator by

and you get:

Hence we get

If ,

then .

If , then must equal 3 (Note that cannot be -3 because you need it to be positive.

Now, plug into the new equation :

43 = 64

Alternatively written, this is 4(4)(4) = 64 or 43 = 641.

Thus, m = 3 and n = 1.

m/n = 3/1 = 3.

The answer is 33 = 27

Since $2^{x}$= $2^{6}$

Hence

From the equation one can see that

Hence must be equal to 25.

Use the rules of logarithms to combine terms.

Hence,

By fatoring we get

Hence .

However, you cannot take the logarithm of a negative number. Thus, the only value for is .

Hence the correct answer is .