Find the Exact Value 12tan(pi/6)^2+24sec(pi/3)^2

$12 \tan^{2} (\frac{π}{6}) + 24 \sec^{2} (\frac{π}{3})$

Step 1

Simplify each term.

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$12 {(\frac{\sqrt{3}}{3})}^{2} + 24 \sec^{2} (\frac{π}{3})$

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

$12 \frac{{\sqrt{3}}^{2}}{3^{2}} + 24 \sec^{2} (\frac{π}{3})$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$12 \frac{{(3^{\frac{1}{2}})}^{2}}{3^{2}} + 24 \sec^{2} (\frac{π}{3})$

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

$12 \frac{3^{\frac{1}{2} \cdot 2}}{3^{2}} + 24 \sec^{2} (\frac{π}{3})$

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

$12 \frac{3^{\frac{2}{2}}}{3^{2}} + 24 \sec^{2} (\frac{π}{3})$

Cancel the common factor of $2$ .

Cancel the common factor.

$12 \frac{3^{\frac{2}{2}}}{3^{2}} + 24 \sec^{2} (\frac{π}{3})$

Rewrite the expression.

$12 \frac{3^{1}}{3^{2}} + 24 \sec^{2} (\frac{π}{3})$

Evaluate the exponent.

$12 \frac{3}{3^{2}} + 24 \sec^{2} (\frac{π}{3})$

Raise $3$ to the power of $2$ .

$12 (\frac{3}{9}) + 24 \sec^{2} (\frac{π}{3})$

Cancel the common factor of $3$ .

Factor $3$ out of $12$ .

$3 (4) \frac{3}{9} + 24 \sec^{2} (\frac{π}{3})$

Factor $3$ out of $9$ .

$3 \cdot 4 \frac{3}{3 \cdot 3} + 24 \sec^{2} (\frac{π}{3})$

Cancel the common factor.

$3 \cdot 4 \frac{3}{3 \cdot 3} + 24 \sec^{2} (\frac{π}{3})$

Rewrite the expression.

$4 (\frac{3}{3}) + 24 \sec^{2} (\frac{π}{3})$

Combine $4$ and $\frac{3}{3}$ .

$\frac{4 \cdot 3}{3} + 24 \sec^{2} (\frac{π}{3})$

Multiply $4$ by $3$ .

$\frac{12}{3} + 24 \sec^{2} (\frac{π}{3})$

Divide $12$ by $3$ .

$4 + 24 \sec^{2} (\frac{π}{3})$

The exact value of $\sec (\frac{π}{3})$ is $2$ .

$4 + 24 \cdot 2^{2}$

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

$4 + 24 \cdot 4$

Multiply $24$ by $4$ .

$4 + 96$

Step 2

Add $4$ and $96$ .

$100$

Find the Exact Value 12tan(pi/6)^2+24sec(pi/3)^2

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