Square root of a product

To solve this problem, we can use the product property of square roots.

• Step 1: Recognize the expression $\sqrt{a} \cdot \sqrt{b}$.
• Step 2: Apply the product property: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$.

This tells us that the original expression, $\sqrt{a} \cdot \sqrt{b}$, simplifies to $\sqrt{a \cdot b}$.

Thus, the equivalent expression is $\sqrt{a \cdot b}$.

Among the given choices, choice 2 $\sqrt{a\cdot b}$ is the correct one.

Let's start with a reminder of the definition of a root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

We will then use the fact that raising the number 1 to any power always yields the result 1, particularly raising it to the power of half of the square root (which we obtain by using the definition of a root as a power mentioned earlier).

In other words:

$$$\sqrt{30}\cdot\sqrt{1}= \\ \downarrow\\ \sqrt{30}\cdot\sqrt[2]{1}=\\ \sqrt{30}\cdot 1^{\frac{1}{2}}=\\ \sqrt{30} \cdot1=\\ \boxed{\sqrt{30}}$

Therefore, the correct answer is answer C.

To solve the expression $\sqrt{1} \cdot \sqrt{25}$, we will use the Product Property of Square Roots.

According to the property, we have:

$\sqrt{1} \cdot \sqrt{25} = \sqrt{1 \cdot 25}$

First, calculate the product inside the square root:

$1 \cdot 25 = 25$

Now the expression simplifies to:

$\sqrt{25}$

Finding the square root of 25 gives us:

$5$

Thus, the value of $\sqrt{1} \cdot \sqrt{25}$ is $\boxed{5}$.

After comparing this solution with the provided choices, we see that the correct answer is choice 3.

Let's start by recalling how to define a root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

Next, we will remember that raising 1 to any power will always yield the result 1, even the half power of the square root.

In other words:

$$$\sqrt{16}\cdot\sqrt{1}= \\ \downarrow\\ \sqrt{16}\cdot\sqrt[2]{1}=\\ \sqrt{16}\cdot 1^{\frac{1}{2}}=\\ \sqrt{16} \cdot1=\\ \sqrt{16} =\\ \boxed{4}$Therefore, the correct answer is answer D.

Let's start by recalling how to define a square root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

Next, we remember that raising 1 to any power always gives us 1, even the half power we got from converting the square root.

In other words:

$$$\sqrt{1} \cdot \sqrt{2}= \\ \downarrow\\ \sqrt[2]{1}\cdot \sqrt{2}=\\ 1^{\frac{1}{2}} \cdot\sqrt{2} =\\ 1\cdot\sqrt{2}=\\ \boxed{\sqrt{2}}$Therefore, the correct answer is answer a.