Rules of Roots - Examples, Exercises and Solutions

Let's begin by calculating the numerical value of each of the roots in the given options:

$\sqrt{25}=5\\ \sqrt{16}=4\\ \sqrt{9}=3\\$We can determine that:

5>4>3>1 Therefore, the correct answer is option A

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 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.

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.

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.

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$b. The law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Let's start by converting the square roots to exponents using the law mentioned in a:

$\sqrt{7}\cdot\sqrt{7}= \\ \downarrow\\ 7^{\frac{1}{2}}\cdot7^{\frac{1}{2}}=$We'll continue, since we are multiplying two terms with identical bases - we'll use the law of exponents mentioned in b:

$7^{\frac{1}{2}}\cdot7^{\frac{1}{2}}= \\ 7^{\frac{1}{2}+\frac{1}{2}}=\\ 7^1=\\ \boxed{7}$Therefore, the correct answer is answer a.

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.

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.

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

$\sqrt{9}\cdot\sqrt{4}=\\ 3\cdot2=\\ \boxed{6}$Therefore, the correct answer is answer B.

Let's simplify the expression. First, we'll reduce the fraction under the square root, then we'll calculate the result of the root:

$\sqrt{\frac{225}{25}}= \\ \sqrt{9}\\ \boxed{3}$Therefore, the correct answer is option B.

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$b. The law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Let's start by converting the square roots to exponents using the law mentioned in a:

$\sqrt{5}\cdot\sqrt{5}= \\ \downarrow\\ 5^{\frac{1}{2}}\cdot5^{\frac{1}{2}}=$We'll continue, since we are multiplying two terms with identical bases - we'll use the law of exponents mentioned in b:

$5^{\frac{1}{2}}\cdot5^{\frac{1}{2}}= \\ 5^{\frac{1}{2}+\frac{1}{2}}=\\ 5^1=\\ \boxed{5}$Therefore, the correct answer is answer a.

In order 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 bases (in the opposite direction):

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

Let's start by changing the square roots to exponents using the law of exponents shown in A:

$\sqrt{2}\cdot\sqrt{5}= \\ \downarrow\\ 2^{\frac{1}{2}}\cdot5^{\frac{1}{2}}=$We continue: since we are multiplying two terms with equal exponents we can use the law of exponents shown in B and combine them together as the same base raised to the same power:

$2^{\frac{1}{2}}\cdot5^{\frac{1}{2}}= \\ (2\cdot5)^{\frac{1}{2}}=\\ 10^{\frac{1}{2}}=\\ \boxed{\sqrt{10}}$In the last steps wemultiplied the bases and then 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 answer B.

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 multiplying exponents for identical bases:

$(a^m)^n=a^{m\cdot n}$

Let's start from the square root of the exponents using the law shown in A:

$\sqrt{2}\cdot\sqrt{2}= \\ \downarrow\\ 2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}=$We continue: note that we got a number times itself. According to the definition of the exponent we can write the expression as an exponent of that number. Then- we use the law of exponents shown in B and perform the whole exponent on the term in the parentheses:

$2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}= \\ (2^{\frac{1}{2}})^2=\\ 2^{\frac{1}{2}\cdot2}=\\ 2^1=\\ \boxed{2}$Therefore, the correct answer is answer B.

Although the square root of 9 is known (3) , in order to get a single expression we will use the laws of parentheses:

So- in order to simplify the given expression we will use two exponents laws:

A. Defining the root as a an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$B. Multiplying different bases with the same power (in the opposite direction):

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

Let's start by changing the square root into an exponent using the law shown in A:

$\sqrt{9}\cdot\sqrt{3}= \\ \downarrow\\ 9^{\frac{1}{2}}\cdot3^{\frac{1}{2}}=$Since a multiplication is performed between two bases with the same exponent we can use the law shown in B.

$9^{\frac{1}{2}}\cdot3^{\frac{1}{2}}= \\ (9\cdot3)^{\frac{1}{2}}=\\ 27^{\frac{1}{2}}=\\ \boxed{\sqrt{27}}$In the last steps we performed the multiplication, and then used the law of defining the root as an exponent shown earlier in A (in the opposite direction) in order to return to the root notation.

Therefore, the correct answer is answer C.

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$b. The law of exponents for power of a power:

$(a^m)^n=a^{m\cdot n}$

Let's start with converting the square root to an exponent using the law mentioned in a':

$\sqrt{x^2}= \\ \downarrow\\ (x^2)^{\frac{1}{2}}=$We'll continue using the law of exponents mentioned in b' and perform the exponent operation on the term in parentheses:

$(x^2)^{\frac{1}{2}}= \\ x^{2\cdot\frac{1}{2}}\\ x^1=\\ \boxed{x}$Therefore, the correct answer is answer a'.