» » » Evaluate $$\sqrt{15} \times (\sqrt{3})^3$$

# Evaluate $$\sqrt{15} \times (\sqrt{3})^3$$

### Question

Evaluate $$\sqrt{15} \times (\sqrt{3})^3$$

### Options

A)
$$15\sqrt{3}$$
B)
$$3\sqrt{15}$$
C)
$$9\sqrt{5}$$
D)
$$\sqrt{45}$$

### Explanation:

To evaluate this expression, $$\sqrt{15} \times (\sqrt{3})^3$$, we'll first simplify $$(\sqrt{3})^3$$ and then multiply the results.

Recall that $$(\sqrt{a})^b = a^{b/2}$$. In this case, $$a = 3$$ and $$b = 3$$, so we have:

$(\sqrt{3})^3 = 3^{3/2}$

Now, we can rewrite the exponent as a product of two fractions:

$3^{3/2} = 3^{1 \times (3/2)} = (3^1)^{3/2}$

We know that $$3^1 = 3$$, so the expression simplifies to:

$(3^1)^{3/2} = 3^{3/2}$

Next, let's multiply $$\sqrt{15}$$ by $$3^{3/2}$$:

$\sqrt{15} \times 3^{3/2}$

Since we're multiplying two square roots, we can combine them using the product property of square roots, which states that $$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$. In this case, $$a = 15$$ and $$b = 3^{3/2}$$, so we have:

$\sqrt{15} \times 3^{3/2} = \sqrt{15 \times 3^{3/2}}$

The expression inside the square root can be simplified as:

$15 \times 3^{3/2} = 15 \times 3 \times \sqrt{3} = 45 \times \sqrt{3}$

Finally, we have:

$\sqrt{15} \times (\sqrt{3})^3 = \sqrt{45 \times \sqrt{3}}$

Option C, $$9\sqrt{5}$$, is the correct answer. To see this, we can rewrite the expression as follows:

$\sqrt{45 \times \sqrt{3}} = \sqrt{9 \times 5 \times \sqrt{3}} = \sqrt{(9 \times \sqrt{3}) \times 5} = 9\sqrt{5}$

## Dicussion (1)

• √15×(√3)$$^3$$
√15×√27 (√3×√3×√3 = √27)
√15×√9×3 = √15×√9×√3
= √5×√3×3×√3
= √5×√3×√3×3
= √5×√9×√3
= √5×3×3
= √5×9
= 9√5.