Expand the Trigonometric Expression ( square root of c-1)^6

${(\sqrt{c - 1})}^{6}$

Step 1

Rewrite ${\sqrt{c - 1}}^{6}$ as ${(c - 1)}^{3}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{c - 1}$ as ${(c - 1)}^{\frac{1}{2}}$ .

${({(c - 1)}^{\frac{1}{2}})}^{6}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(c - 1)}^{\frac{1}{2} \cdot 6}$

Combine $\frac{1}{2}$ and $6$ .

${(c - 1)}^{\frac{6}{2}}$

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

${(c - 1)}^{\frac{2 \cdot 3}{2}}$

Cancel the common factors.

Factor $2$ out of $2$ .

${(c - 1)}^{\frac{2 \cdot 3}{2 (1)}}$

Cancel the common factor.

${(c - 1)}^{\frac{2 \cdot 3}{2 \cdot 1}}$

Rewrite the expression.

${(c - 1)}^{\frac{3}{1}}$

Divide $3$ by $1$ .

${(c - 1)}^{3}$

Step 2

Use the Binomial Theorem.

$c^{3} + 3 c^{2} \cdot - 1 + 3 c {(- 1)}^{2} + {(- 1)}^{3}$

Step 3

Simplify each term.

Multiply $- 1$ by $3$ .

$c^{3} - 3 c^{2} + 3 c {(- 1)}^{2} + {(- 1)}^{3}$

Raise $- 1$ to the power of $2$ .

$c^{3} - 3 c^{2} + 3 c \cdot 1 + {(- 1)}^{3}$

Multiply $3$ by $1$ .

$c^{3} - 3 c^{2} + 3 c + {(- 1)}^{3}$

Raise $- 1$ to the power of $3$ .

$c^{3} - 3 c^{2} + 3 c - 1$