Square root of a product

Solve the following exercise:

$\sqrt{16}\cdot\sqrt{1}=$

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.

$4$

Solve the following exercise:

$\sqrt{1}\cdot\sqrt{2}=$

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.

$\sqrt{2}$

Solve the following exercise:

$\sqrt{10}\cdot\sqrt{3}=$

To simplify the given expression, we use two laws of exponents:

A. Defining the root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$B. The law of exponents for dividing powers with the same base (in the opposite direction):

$x^n\cdot y^n =(x\cdot y)^n$

Let's start by using the law of exponents shown in A:

$\sqrt{10}\cdot\sqrt{3}= \\ \downarrow\\ 10^{\frac{1}{2}}\cdot3^{\frac{1}{2}}=$We continue, since we have a multiplication between two terms with equal exponents, we can use the law of exponents shown in B and combine them under the same base which is raised to the same exponent:

$10^{\frac{1}{2}}\cdot3^{\frac{1}{2}}= \\ (10\cdot3)^{\frac{1}{2}}=\\ 30^{\frac{1}{2}}=\\ \boxed{\sqrt{30}}$In the last steps, we performed the multiplication of the bases and used the definition of the root as an exponent shown earlier in A (in the opposite direction) to return to the root notation.

Therefore, the correct answer is B.

$\sqrt{30}$

Solve the following exercise:

$\sqrt{100}\cdot\sqrt{25}=$

We can simplify the expression without using the laws of exponents, because the expression has known square roots, so let's simplify the expression and then perform the multiplication:

$\sqrt{100}\cdot\sqrt{25}=\\ 10\cdot5=\\ \boxed{50}$Therefore, the correct answer is answer D.

$50$

Solve the following exercise:

$\sqrt{25}\cdot\sqrt{4}=$

We can simplify the expression directly without using the laws of exponents, since the expression has known square roots, so let's simplify the expression and then perform the multiplication:

$\sqrt{25}\cdot\sqrt{4}=\\ 5\cdot2=\\ \boxed{10}$Therefore, the correct answer is answer C.

$10$