Rules of Roots Combined: Using variables

To simplify the given expression, we will use 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:

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

We'll start 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.

To simplify the given expression, we will use 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:

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

We'll start by converting the fourth root to an exponent using the law of exponents mentioned in a.:

$\sqrt{100x^2}= \\ \downarrow\\ (100x^2)^{\frac{1}{2}}=$

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

$(100x^2)^{\frac{1}{2}}= \\ 100^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}}$

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):

$100^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}} = \\ 100^{\frac{1}{2}}\cdot x^{2\cdot\frac{1}{2}}=\\ 100^{\frac{1}{2}}\cdot x^{1}=\\ \sqrt{100}\cdot x=\\ \boxed{10x}$

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 reverse) and then calculated the known fourth root of 100.

Therefore, the correct answer is answer d.

To simplify the given expression, we will use 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:

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

We'll start with converting the fourth root to an exponent using the law of exponents mentioned in a.:

$\sqrt{16x^2}= \\ \downarrow\\ (16x^2)^{\frac{1}{2}}=$

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

$(16x^2)^{\frac{1}{2}}= \\ 16^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}}$

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):

$16^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}} = \\ 16^{\frac{1}{2}}\cdot x^{2\cdot\frac{1}{2}}=\\ 16^{\frac{1}{2}}\cdot x^{1}=\\ \sqrt{16}\cdot x=\\ \boxed{4x}$

In the final steps, first we 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 opposite direction) and then we calculated the known fourth root of 16.

Therefore, the correct answer is answer d.

To simplify the given expression, we will use 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:

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

We'll start by converting the fourth root to an exponent using the law of exponents mentioned in a.:

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

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

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

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):

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

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.

In order to simplify the given expression, we will use the following 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 an exponent applied to terms in parentheses:

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

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

$\sqrt{36x}= \\ \downarrow\\ (36x)^{\frac{1}{2}}=$

Next, we'll use the law of exponents mentioned in b and apply the exponent to each factor within the parentheses:

$(36x)^{\frac{1}{2}}= \\ 36^{\frac{1}{2}}\cdot x^{{\frac{1}{2}}}=\\ \sqrt{36}\sqrt{x}=\\ \boxed{6\sqrt{x}}$$$

In the final steps, we first converted the power of one-half applied to each factor in the multiplication back to square root form, again, according to the definition of root as an exponent mentioned in a (in the opposite direction) and then calculated the known square root of 36.

Therefore, the correct answer is answer c.

To simplify the given expression, we will use 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:

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

We'll start with converting the fourth root to an exponent using the law of exponents mentioned in a.:

$\sqrt{49x^2}= \\ \downarrow\\ (49x^2)^{\frac{1}{2}}=$

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

$(49x^2)^{\frac{1}{2}}= \\ 49^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}}$

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):

$49^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}} = \\ 49^{\frac{1}{2}}\cdot x^{2\cdot\frac{1}{2}}=\\ 49^{\frac{1}{2}}\cdot x^{1}=\\ \sqrt{49}\cdot x=\\ \boxed{7x}$

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 reverse) and then calculated the known fourth root of 49.

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'.

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

a. The definition of root as an exponent:

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

b. Law of exponents for power to a power:

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

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

$\sqrt{x^4}= \\ \downarrow\\ (x^4)^{\frac{1}{2}}=$Let's continue, using the law of exponents mentioned in b to perform the exponentiation of the term in parentheses:

$(x^4)^{{\frac{1}{2}}} = \\ x^{4\cdot\frac{1}{2}}=\\ \boxed{x^2}$Therefore, the correct answer is answer b.

In order to simplify the given expression, we will use 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:

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

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

$\sqrt{11x^2}= \\ \downarrow\\ (11x^2)^{\frac{1}{2}}=$

Next, we'll use the law of exponents mentioned in b and apply the exponent to each term in the parentheses:

$(11x^2)^{\frac{1}{2}}= \\ 11^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}}$

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

$11^{\frac{1}{2}}\cdot(x^2)^{{\frac{1}{2}}} = \\ 11^{\frac{1}{2}}\cdot x^{2\cdot\frac{1}{2}}=\\ 11^{\frac{1}{2}}\cdot x^{1}=\\ \boxed{\sqrt{11} x}\\$In the final step, we converted the one-half exponent applied to the first term in the multiplication back to a fourth root, again, according to the definition of root as an exponent mentioned in a (in the reverse direction).

Therefore, the correct answer is answer c.

To simplify the given expression, we will use 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:

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

Let's start with converting the fourth root to an exponent using the law of exponents mentioned in a.:

$\sqrt{12x^4}= \\ \downarrow\\ (12x^4)^{\frac{1}{2}}=$

Next, we'll use the law of exponents mentioned in b. and apply the exponent to each factor in the parentheses:

$(12x^4)^{\frac{1}{2}}= \\ 12^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}}$

Let's continue, using the law of exponents mentioned in c. and perform the exponent operation on the term with an exponent in the parentheses (the second term in the product):

$12^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}} = \\ 12^{\frac{1}{2}}\cdot x^{4\cdot\frac{1}{2}}=\\ 12^{\frac{1}{2}}\cdot x^{2}=\\ \boxed{\sqrt{12} x^2}=\\$In the final step, we converted the one-half exponent on the first term in the product back to a fourth root form, again, according to the definition of root as an exponent mentioned in a. (in the reverse direction).

Therefore, the correct answer is answer c.

To simplify the given expression, we will use 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:

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

We'll start by converting the fourth root to an exponent using the law of exponents mentioned in a.:

$\sqrt{36x^4}= \\ \downarrow\\ (36x^4)^{\frac{1}{2}}=$

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

$(36x^4)^{\frac{1}{2}}= \\ 36^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}}$

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):

$36^{\frac{1}{2}}\cdot(x^4)^{{\frac{1}{2}}} = \\ 36^{\frac{1}{2}}\cdot x^{4\cdot\frac{1}{2}}=\\ 36^{\frac{1}{2}}\cdot x^{2}=\\ \sqrt{36}\cdot x^2=\\ \boxed{6x^2}$

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 36.

Therefore, the correct answer is answer d.

To simplify the given expression, we will use 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:

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

We'll start with converting the fourth root to an exponent using the law of exponents mentioned in a.:

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

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

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

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):

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

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 reverse) and then calculated the known fourth root of 4.

Therefore, the correct answer is answer b.

To simplify the given expression, we will use 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:

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

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

$\sqrt{5x^4}= \\ \downarrow\\ (5x^4)^{\frac{1}{2}}=$

Next, we'll use the law of exponents mentioned in b and apply the exponent to each factor in the parentheses:

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

Let's continue, using the law of exponents mentioned in c and perform the exponent operation on the term with an exponent in the parentheses (the second term in the multiplication):

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

In the final step, we converted the one-half power applied to the first term 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).

Therefore, the correct answer is answer c.

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

A. Definition of the root as an exponent:

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

B. Law of exponents for dividing powers with the same base:

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

Let's start with converting the root to an exponent using the law of exponents shown in A:

$\sqrt{9x}= \\ \downarrow\\ (9x)^{\frac{1}{2}}=$Next, we will use the law of exponents shown in B and apply the exponent to each of the factors in the numerator that are in parentheses:

$(9x)^{\frac{1}{2}}= \\ 9^{\frac{1}{2}}\cdot x^{{\frac{1}{2}}}=\\ \sqrt{9}\sqrt{x}=\\ \boxed{3\sqrt{x}}$$$In the last steps, we will multiply the half exponent by each of the factors in the numerator, returning to the root form, that is, according to the definition of the root as an exponent shown in A (in the opposite direction) and then we will calculate the known fourth root result of the number 9.

Therefore, the correct answer is answer D.