Simplify the Radical Expression ( square root of 3t^2s^3)/(4 square root of 5ts^3)

$\frac{\sqrt{3 t^{2} s^{3}}}{4 \sqrt{5 t s^{3}}}$

Step 1

Combine $\sqrt{3 t^{2} s^{3}}$ and $\sqrt{5 t s^{3}}$ into a single radical.

$\frac{\sqrt{\frac{3 t^{2} s^{3}}{5 t s^{3}}}}{4}$

Step 2

Cancel the common factor of $t^{2}$ and $t$ .

Factor $t$ out of $3 t^{2} s^{3}$ .

$\frac{\sqrt{\frac{t (3 t s^{3})}{5 t s^{3}}}}{4}$

Cancel the common factors.

Factor $t$ out of $5 t s^{3}$ .

$\frac{\sqrt{\frac{t (3 t s^{3})}{t (5 s^{3})}}}{4}$

Cancel the common factor.

$\frac{\sqrt{\frac{t (3 t s^{3})}{t (5 s^{3})}}}{4}$

Rewrite the expression.

$\frac{\sqrt{\frac{3 t s^{3}}{5 s^{3}}}}{4}$

Step 3

Cancel the common factor of $s^{3}$ .

Cancel the common factor.

$\frac{\sqrt{\frac{3 t s^{3}}{5 s^{3}}}}{4}$

Rewrite the expression.

$\frac{\sqrt{\frac{3 t}{5}}}{4}$

Step 4

Simplify the numerator.

Rewrite $\sqrt{\frac{3 t}{5}}$ as $\frac{\sqrt{3 t}}{\sqrt{5}}$ .

$\frac{\frac{\sqrt{3 t}}{\sqrt{5}}}{4}$

Multiply $\frac{\sqrt{3 t}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\frac{\sqrt{3 t}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}}{4}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3 t}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\frac{\sqrt{3 t} \sqrt{5}}{\sqrt{5} \sqrt{5}}}{4}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\frac{\sqrt{3 t} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}}{4}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\frac{\sqrt{3 t} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}}{4}$

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

$\frac{\frac{\sqrt{3 t} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}}{4}$

Add $1$ and $1$ .

$\frac{\frac{\sqrt{3 t} \sqrt{5}}{{\sqrt{5}}^{2}}}{4}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\frac{\sqrt{3 t} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}}{4}$

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

$\frac{\frac{\sqrt{3 t} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}}{4}$

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

$\frac{\frac{\sqrt{3 t} \sqrt{5}}{5^{\frac{2}{2}}}}{4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{\sqrt{3 t} \sqrt{5}}{5^{\frac{2}{2}}}}{4}$

Rewrite the expression.

$\frac{\frac{\sqrt{3 t} \sqrt{5}}{5^{1}}}{4}$

Evaluate the exponent.

$\frac{\frac{\sqrt{3 t} \sqrt{5}}{5}}{4}$

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{3 t \cdot 5}}{5}}{4}$

Multiply $5$ by $3$ .

$\frac{\frac{\sqrt{15 t}}{5}}{4}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{15 t}}{5} \cdot \frac{1}{4}$

Step 6

Multiply $\frac{\sqrt{15 t}}{5} \cdot \frac{1}{4}$ .

Multiply $\frac{\sqrt{15 t}}{5}$ by $\frac{1}{4}$ .

$\frac{\sqrt{15 t}}{5 \cdot 4}$

Multiply $5$ by $4$ .

$\frac{\sqrt{15 t}}{20}$

Simplify the Radical Expression ( square root of 3t^2s^3)/(4 square root of 5ts^3)

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