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

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

Step 1

Simplify each term.

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

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

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

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

$12 {(3^{\frac{1}{2}})}^{2} + 24 \sec^{2} (\frac{π}{6})$

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

$12 \cdot 3^{\frac{1}{2} \cdot 2} + 24 \sec^{2} (\frac{π}{6})$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$12 \cdot 3^{1} + 24 \sec^{2} (\frac{π}{6})$

Evaluate the exponent.

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

Multiply $12$ by $3$ .

$36 + 24 \sec^{2} (\frac{π}{6})$

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$36 + 24 {(\frac{2}{\sqrt{3}})}^{2}$

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$36 + 24 {(\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}^{2}$

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$36 + 24 {(\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})}^{2}$

Raise $\sqrt{3}$ to the power of $1$ .

$36 + 24 {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})}^{2}$

Raise $\sqrt{3}$ to the power of $1$ .

$36 + 24 {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$36 + 24 {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}})}^{2}$

Add $1$ and $1$ .

$36 + 24 {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}})}^{2}$

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

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

$36 + 24 {(\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})}^{2}$

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

$36 + 24 {(\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})}^{2}$

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

$36 + 24 {(\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 + 24 {(\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})}^{2}$

Rewrite the expression.

$36 + 24 {(\frac{2 \sqrt{3}}{3^{1}})}^{2}$

Evaluate the exponent.

$36 + 24 {(\frac{2 \sqrt{3}}{3})}^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

$36 + 24 \frac{{(2 \sqrt{3})}^{2}}{3^{2}}$

Apply the product rule to $2 \sqrt{3}$ .

$36 + 24 \frac{2^{2} {\sqrt{3}}^{2}}{3^{2}}$

Simplify the numerator.

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

$36 + 24 \frac{4 {\sqrt{3}}^{2}}{3^{2}}$

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

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

$36 + 24 \frac{4 {(3^{\frac{1}{2}})}^{2}}{3^{2}}$

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

$36 + 24 \frac{4 \cdot 3^{\frac{1}{2} \cdot 2}}{3^{2}}$

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

$36 + 24 \frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 + 24 \frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}}$

Rewrite the expression.

$36 + 24 \frac{4 \cdot 3^{1}}{3^{2}}$

Evaluate the exponent.

$36 + 24 \frac{4 \cdot 3}{3^{2}}$

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

$36 + 24 \frac{4 \cdot 3}{9}$

Multiply $4$ by $3$ .

$36 + 24 (\frac{12}{9})$

Cancel the common factor of $3$ .

Factor $3$ out of $24$ .

$36 + 3 (8) \frac{12}{9}$

Factor $3$ out of $9$ .

$36 + 3 \cdot 8 \frac{12}{3 \cdot 3}$

Cancel the common factor.

$36 + 3 \cdot 8 \frac{12}{3 \cdot 3}$

Rewrite the expression.

$36 + 8 (\frac{12}{3})$

Combine $8$ and $\frac{12}{3}$ .

$36 + \frac{8 \cdot 12}{3}$

Multiply $8$ by $12$ .

$36 + \frac{96}{3}$

Divide $96$ by $3$ .

$36 + 32$

Step 2

Add $36$ and $32$ .

$68$

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

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