Simplify the Radical Expression square root of 3(4+5 square root of 3)

$\sqrt{3} (4 + 5 \sqrt{3})$

Step 1

Apply the distributive property.

$\sqrt{3} \cdot 4 + \sqrt{3} (5 \sqrt{3})$

Step 2

Move $4$ to the left of $\sqrt{3}$ .

$4 \cdot \sqrt{3} + \sqrt{3} (5 \sqrt{3})$

Step 3

Multiply $\sqrt{3} (5 \sqrt{3})$ .

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

$4 \cdot \sqrt{3} + 5 ({\sqrt{3}}^{1} \sqrt{3})$

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

$4 \cdot \sqrt{3} + 5 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

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

$4 \cdot \sqrt{3} + 5 {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

$4 \cdot \sqrt{3} + 5 {\sqrt{3}}^{2}$

Step 4

Simplify each term.

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$4 \sqrt{3} + 5 \cdot 3^{1}$

Evaluate the exponent.

$4 \sqrt{3} + 5 \cdot 3$

Multiply $5$ by $3$ .

$4 \sqrt{3} + 15$

Step 5

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{3} + 15$

Decimal Form:

$21.92820323 \dots$