Rules of Roots Combined: Using multiple rules

Identify the greater valueSame base and different indicatorUsing variablesSolving the equationApplying the formulaNumber of terms
Question 1

Solve the following exercise:

\( \sqrt[4]{\sqrt[3]{3}}= \)

Question 2

Solve the following exercise:

\( \frac{\sqrt{20}\cdot\sqrt{4}}{\sqrt{5}}= \)

Question 3

Solve the following exercise:

\( \frac{2^3\cdot2^4}{2^5}= \)

Question 4

Solve the following exercise:

\( \frac{\sqrt{100}}{\sqrt{25}\cdot\sqrt{4}}= \)

Question 5

Solve the following exercise:

\( \sqrt[3]{\sqrt{64}}\cdot\sqrt{64}= \)

Examples with solutions for Rules of Roots Combined: Using multiple rules

Exercise #1

Solve the following exercise:

334= \sqrt[4]{\sqrt[3]{3}}=

Video Solution

Step-by-Step Solution

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

A. Definition of the root as an exponent:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}}

B. Law of exponents for an exponent on an exponent:

(am)n=am⋅n (a^m)^n=a^{m\cdot n}

Let's begin simplifying the given expression:

334= \sqrt[4]{\sqrt[3]{3}}= \\ We will use the law of exponents shown in A and first convert the roots in the expression to exponents, we will do this in two steps - in the first step we will convert the inner root in the expression and in the next step we will convert the outer root:

334=3134=(313)14= \sqrt[4]{\sqrt[3]{3}}= \\ \sqrt[4]{3^{\frac{1}{3}}}= \\ (3^{\frac{1}{3}})^{\frac{1}{4}}= We continue and use the law of exponents shown in B, then we will multiply the exponents:

(313)14=313⋅14=31⋅13⋅4=3112=312 (3^{\frac{1}{3}})^{\frac{1}{4}}= \\ 3^{\frac{1}{3}\cdot\frac{1}{4}}=\\ 3^{\frac{1\cdot1}{3\cdot4}}=\\ \boxed{3^{\frac{1}{12}}}=\\ \boxed{\sqrt[12]{3}} In the final step we return to writing the root, that is - back, using the law of exponents shown in A (in the opposite direction),

Let's summarize the simplification of the given expression:

334=(313)14=3112=312 \sqrt[4]{\sqrt[3]{3}}= \\ (3^{\frac{1}{3}})^{\frac{1}{4}}= \\ \boxed{3^{\frac{1}{12}}}=\\ \boxed{\sqrt[12]{3}} Therefore, note that the correct answer (most) is answer D.

Answer

Answers a + b

Exercise #2

Solve the following exercise:

20⋅45= \frac{\sqrt{20}\cdot\sqrt{4}}{\sqrt{5}}=

Video Solution

Answer

4 4

Exercise #3

Solve the following exercise:

23⋅2425= \frac{2^3\cdot2^4}{2^5}=

Video Solution

Answer

4 4

Exercise #4

Solve the following exercise:

10025⋅4= \frac{\sqrt{100}}{\sqrt{25}\cdot\sqrt{4}}=

Video Solution

Answer

1 1

Exercise #5

Solve the following exercise:

643⋅64= \sqrt[3]{\sqrt{64}}\cdot\sqrt{64}=

Video Solution

Answer

16

Question 1

\( \frac{\sqrt{35}\cdot\sqrt{20}}{\sqrt{7}}= \)

Question 2

Solve the following exercise:

\( \frac{\sqrt{70}\cdot\sqrt{10}}{\sqrt{7}}= \)

Question 3

Solve the following exercise:

\( \frac{\sqrt{2}\cdot\sqrt{9}\cdot\sqrt{2}}{\sqrt{3}\cdot\sqrt{4}}= \)

Question 4

Solve the following exercise:

\( \frac{\sqrt{4}\cdot\sqrt{9}}{\sqrt{16}}= \)

Question 5

Solve the following exercise:

\( \sqrt{\frac{36}{144}}\cdot\sqrt{\sqrt{16}}= \)

Exercise #6

35⋅207= \frac{\sqrt{35}\cdot\sqrt{20}}{\sqrt{7}}=

Video Solution

Answer

10 10

Exercise #7

Solve the following exercise:

70⋅107= \frac{\sqrt{70}\cdot\sqrt{10}}{\sqrt{7}}=

Video Solution

Answer

10 10

Exercise #8

Solve the following exercise:

2⋅9⋅23⋅4= \frac{\sqrt{2}\cdot\sqrt{9}\cdot\sqrt{2}}{\sqrt{3}\cdot\sqrt{4}}=

Video Solution

Answer

3 \sqrt{3}

Exercise #9

Solve the following exercise:

4⋅916= \frac{\sqrt{4}\cdot\sqrt{9}}{\sqrt{16}}=

Video Solution

Answer

32 \frac{3}{2}

Exercise #10

Solve the following exercise:

36144⋅16= \sqrt{\frac{36}{144}}\cdot\sqrt{\sqrt{16}}=

Video Solution

Answer

1

Question 1

Solve the following exercise:

\( \frac{\sqrt{10}\cdot\sqrt{30}}{\sqrt{100}}= \)

Question 2

Solve the following exercise:

\( \frac{\sqrt[4]{128}}{\sqrt[4]{8}}= \)

Question 3

Solve the following exercise:

\( \frac{\sqrt{2}\cdot\sqrt{6}\cdot\sqrt{12}}{\sqrt{16}}= \)

Question 4

Solve the following exercise:

\( \sqrt[7]{\sqrt{5}}\cdot\sqrt[14]{\sqrt{5}}= \)

Question 5

Solve the following exercise:

\( \sqrt{\sqrt{\frac{100}{25}}}\cdot\sqrt{\sqrt{25}}= \)

Exercise #11

Solve the following exercise:

10⋅30100= \frac{\sqrt{10}\cdot\sqrt{30}}{\sqrt{100}}=

Video Solution

Answer

3 \sqrt{3}

Exercise #12

Solve the following exercise:

128484= \frac{\sqrt[4]{128}}{\sqrt[4]{8}}=

Video Solution

Answer

2

Exercise #13

Solve the following exercise:

2⋅6⋅1216= \frac{\sqrt{2}\cdot\sqrt{6}\cdot\sqrt{12}}{\sqrt{16}}=

Video Solution

Answer

3

Exercise #14

Solve the following exercise:

57⋅514= \sqrt[7]{\sqrt{5}}\cdot\sqrt[14]{\sqrt{5}}=

Video Solution

Answer

5114+128 5^{\frac{1}{14}+\frac{1}{28}}

Exercise #15

Solve the following exercise:

10025⋅25= \sqrt{\sqrt{\frac{100}{25}}}\cdot\sqrt{\sqrt{25}}=

Video Solution

Answer

10 \sqrt{10}

Question 1

Solve the following exercise:

\( \sqrt{\frac{16}{\sqrt[3]{64}}}= \)

Exercise #16

Solve the following exercise:

16643= \sqrt{\frac{16}{\sqrt[3]{64}}}=

Video Solution

Answer

2