Rules of Roots Combined - Examples, Exercises and Solutions

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.

In order to simplify the given expression, apply the following three laws of exponents:

a. Definition of root as an exponent:

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

b. Law of exponents for an exponent applied to terms in parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

c. Law of exponents for an exponent raised to an exponent:

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

Begin by converting the fourth root to an exponent using the law of exponents mentioned in a.:

$\sqrt{25x^4}= \\ \downarrow\\ (25x^4)^{\frac{1}{2}}=$

We'll continue, using the law of exponents mentioned in b. and apply the exponent to each factor in the parentheses:

$(25x^4)^{\frac{1}{2}}= \\ 25^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}}$

We'll continue, using the law of exponents mentioned in c. and perform the exponent applied to the term with an exponent in parentheses (the second factor in the multiplication):

$25^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}} = \\ 25^{\frac{1}{2}}\cdot x^{4\cdot\frac{1}{2}}=\\ 25^{\frac{1}{2}}\cdot x^{2}=\\ \sqrt{25}\cdot x^2=\\ \boxed{5x^2}$

In the final steps, we first converted the power of one-half applied to the first factor in the multiplication back to the fourth root form, again, according to the definition of root as an exponent mentioned in a. (in the reverse direction) and then calculated the known fourth root of 25.

Therefore, the correct answer is answer a.

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

Simplify the following expression:

Begin by reducing the fraction under the square root:

$\sqrt{\frac{2}{4}}= \\ \sqrt{\frac{1}{2}}=$

Apply two exponent laws:

A. Definition of root as a power:

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

B. The power law for powers applied to terms in parentheses:

$\big(\frac{a}{b}\big)^n=\frac{a^n}{b^n}$

Let's return to the expression that we obtained. Apply the law mentioned in A and convert the square root to a power:

$\sqrt{\frac{1}{2}}=\\ \big(\frac{1}{2}\big)^{\frac{1}{2}}=$

Next use the power law mentioned in B, apply the power separately to the numerator and denominator.

In the next step remember that raising the number 1 to any power will always result in 1.

In the fraction's denominator we'll return to the root notation, again, using the power law mentioned in A (in the opposite direction):

$\big(\frac{1}{2}\big)^{\frac{1}{2}}= \\ \frac{1^{\frac{1}{2}}}{2^{\frac{1}{2}}}=\\ \boxed{\frac{1}{\sqrt{2}}}\\$ Let's summarize the simplification of the given expression:

$\sqrt{\frac{2}{4}}= \\ \sqrt{\frac{1}{2}}= \\ \frac{1^{\frac{1}{2}}}{2^{\frac{1}{2}}}=\\ \boxed{\frac{1}{\sqrt{2}}}\\$Therefore, the correct answer is answer D.

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.

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

a. The definition of a 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 by converting the square roots to exponents using the law mentioned in a:

$\sqrt{3}\cdot\sqrt{3}= \\ \downarrow\\ 3^{\frac{1}{2}}\cdot3^{\frac{1}{2}}=$Let's continue, notice that we got a number multiplied by itself, therefore, according to the definition of exponents we can write the expression we got as a power of that same number, then we'll use the law of exponents mentioned in b and perform the exponentiation of the term in parentheses:

$3^{\frac{1}{2}}\cdot3^{\frac{1}{2}}= \\ (3^{\frac{1}{2}})^2=\\ 3^{\frac{1}{2}\cdot2}=\\ 3^1=\\ \boxed{3}$Additionally, we identify that:

$3=\sqrt{9}$Therefore, the correct answer (most accurate) is answer d.

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, apply 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 exponents applied to terms in parentheses (in reverse):

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

Begin by converting the square roots to exponents using the law of exponents mentioned in a':

$\sqrt{5}\cdot\sqrt{6}= \\ \downarrow\\ 5^{\frac{1}{2}}\cdot6^{\frac{1}{2}}=$

Due to the fact that there is multiplication between two terms with identical exponents, we are able to apply the law of exponents mentioned in b' and then proceed to combine them together inside of parentheses, raised to the same exponent:

$5^{\frac{1}{2}}\cdot6^{\frac{1}{2}}= \\ (5\cdot6)^{\frac{1}{2}}=\\ 30^{\frac{1}{2}}=\\ \boxed{\sqrt{30}}$

In the final steps, we performed the multiplication within the parentheses and again used the definition of root as an exponent mentioned in a' (in reverse) to return to root notation.

Therefore, the correct answer is answer d.

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.

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

$\sqrt{4}\cdot\sqrt{4}= \\ \downarrow\\ 4^{\frac{1}{2}}\cdot4^{\frac{1}{2}}=$Let's continue, notice that we got a number multiplied by itself, therefore, according to the definition of exponents we can write the expression we got as a power of that same number, then - we'll use the law of exponents mentioned in b' and perform the exponent operation on the term in parentheses:

$4^{\frac{1}{2}}\cdot4^{\frac{1}{2}}= \\ (4^{\frac{1}{2}})^2=\\ 4^{\frac{1}{2}\cdot2}=\\ 4^1=\\ \boxed{4}$Therefore, the correct answer is answer c.

In order to simplify the given expression, apply 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 exponents applied to terms in parentheses (in reverse):

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

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

$\sqrt{2}\cdot\sqrt{3}= \\ \downarrow\\ 2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}=$

Due to the fact that there is multiplication between two terms with identical exponents, we can apply the law of exponents mentioned in b' and then proceed to combine them together inside of parentheses, which are raised to the same exponent:

$2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}= \\ (2\cdot3)^{\frac{1}{2}}=\\ 6^{\frac{1}{2}}=\\ \boxed{\sqrt{6}}$

In the final steps, we performed the multiplication within the parentheses and again used the definition of root as an exponent mentioned earlier in a' (in reverse) to return to root notation.

Therefore, the correct answer is answer b.