How to find a rational number from an exponent - 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 :

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 :

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 .