Simplify indices expression

Question

Neither I nor my maths teacher can simplify this expression to get the answer in our textbook. Can you show us how, or is the textbook wrong?

The expression

The textbook claims the simplified expression is:$$4/21$$

Answer 1

It is simply a matter of rearranging the numerator and denominator:

Observe that your equation is the same as:

$$\frac{7^{2x+1}-7^{2x-1}-7^2+1}{36 \cdot 7^{2x} - 36 \cdot 7}$$

After rearranging numerator and factorizing denominator this simplifies to:

$$\frac{(7^{2x+1}-7^{2})-(7^{2x-1}-1)}{36 \cdot 7(7^{2x-1} - 1)}$$

More factorizing on numerator

$$\frac{7^2(7^{2x-1}-1)-(7^{2x-1}-1)}{36 \cdot 7(7^{2x-1} - 1)}$$

Now it's just $$\frac{49-1}{252}=\frac{4}{21}$$

It would seem your textbook is right.