Simplify arcsin( square root of 3/2)

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Trigonometry

$\arcsin (\sqrt{\frac{3}{2}})$

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

$\arcsin (\frac{\sqrt{3}}{\sqrt{2}})$

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

$\arcsin (\frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Combine and simplify the denominator.

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

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Raise $\sqrt{2}$ to the power of $1$ .

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})$

Raise $\sqrt{2}$ to the power of $1$ .

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})$

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

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

Add $1$ and $1$ .

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}})$

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

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Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})$

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

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})$

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

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}})$

Divide $1$ by $1$ .

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{2^{1}})$

Evaluate the exponent.

$\arcsin (\frac{\sqrt{3} \sqrt{2}}{2})$

Simplify the numerator.

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Combine using the product rule for radicals.

$\arcsin (\frac{\sqrt{3 \cdot 2}}{2})$

Multiply $3$ by $2$ .

$\arcsin (\frac{\sqrt{6}}{2})$

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