Factor -c^10 n^6 + 1/49.

Step-by-step solution

1. 1

   Reorder so the positive term comes first. A difference of squares is easiest to spot in the form $A^2-B^2$:

   $-c^{10}n^6+\frac{1}{49}=\frac{1}{49}-c^{10}n^6$

2. 2

   Express each term as a square. For the fraction, take the square root of numerator and denominator; for the monomial, halve each exponent:

   $\frac{1}{49}=\left(\frac{1}{7}\right)^2,\qquad c^{10}n^6=\left(c^5n^3\right)^2$

   Halving works because $\left(c^5\right)^2=c^{10}$ and $\left(n^3\right)^2=n^6$; both exponents being even is exactly what makes this a square.

3. 3

   Apply the identity $A^2-B^2=(A-B)(A+B)$. With $A=\tfrac17$ and $B=c^5n^3$:

   $\frac{1}{49}-c^{10}n^6=\left(\frac{1}{7}-c^5n^3\right)\left(\frac{1}{7}+c^5n^3\right)$

4. 4

   Confirm no further factorisation is available. Neither factor is a difference of squares (one has a plus sign, and the other has an odd power $c^5$), and neither has a common factor. Check numerically with $c=n=1$: original $=\tfrac1{49}-1=-\tfrac{48}{49}$, and $\left(\tfrac17-1\right)\left(\tfrac17+1\right)=\left(-\tfrac67\right)\left(\tfrac87\right)=-\tfrac{48}{49}$ $\checkmark$.