Rules of Roots Combined - Examples, Exercises and Solutions

Understanding Rules of Roots Combined

Complete explanation with examples

Combining root laws

What is a root?

A root is the inverse operation of exponentiation, denoted by the symbol , and it is equivalent to the power of 0.50.5.
If a small number appears on the left side, it indicates the order of the root.

Things to know about roots:

Square root of a product

(ab)=ab\sqrt{(a\cdot b)}=\sqrt{a}\cdot\sqrt{b}

Square root of a quotient

ab=ab\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}

Square root of a square root

amn=anm\sqrt[n]{\sqrt[m]{a}}=\sqrt[n\cdot m]{a}

Practice

The following exercise combines all the rules of roots,
can you solve it?

416+64327+101+3=\sqrt{4\cdot16}+\sqrt{\frac{64}{3\cdot27}}+\sqrt{10-1}+3=

Solution:
Roots come before the order of operations, so we will first deal with the first root:
416=416\sqrt{4\cdot16}=\sqrt4\cdot\sqrt{16}
We could do this using the root formula of a product.
Let's move on to the second root:
64327=6481=6481\sqrt{\frac{64}{3\cdot27}}=\sqrt{\frac{64}{81}}=\frac{\sqrt{64}}{\sqrt{81}}
Note – in this root, there was an exercise in the denominator, we first solved it and then continued to simplify the root using the root formula of a quotient.
Let's move on to the third root:
101=9\sqrt{10-1}=\sqrt9
Here we simply solved the exercise inside the root without using a formula.
Now let's rewrite the exercise slowly and carefully without getting confused:
416+64819+3=\sqrt4\cdot\sqrt{16}+\frac{\sqrt{64}}{\sqrt{81}}\cdot\sqrt9+3=
There are still roots in the exercise, so we will need to get rid of them:
24+893+3=2\cdot4+\frac{8}{9}\cdot3+3=
Now that there are no more roots, we can solve according to the order of operations:
8+223+3=13238+2\frac{2}{3}+3=13\dfrac{2}{3}

Detailed explanation

Practice Rules of Roots Combined

Test your knowledge with 25 quizzes

Solve the following exercise:

\( \sqrt{100x^2}= \)

Examples with solutions for Rules of Roots Combined

Step-by-step solutions included
Exercise #1

Solve the following exercise:

25x4= \sqrt{25x^4}=

Step-by-Step Solution

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

a. Definition of root as an exponent:

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

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

(ab)n=anbn (a\cdot b)^n=a^n\cdot b^n

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

(am)n=amn (a^m)^n=a^{m\cdot n}

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

25x4=(25x4)12= \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:

(25x4)12=2512(x4)12 (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):

2512(x4)12=2512x412=2512x2=25x2=5x2 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.

Answer:

5x2 5x^2

Video Solution
Exercise #2

Choose the largest value

Step-by-Step Solution

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

25=516=49=3 \sqrt{25}=5\\ \sqrt{16}=4\\ \sqrt{9}=3\\ We can determine that:

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

Answer:

25 \sqrt{25}

Video Solution
Exercise #3

Solve the following exercise:

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

Step-by-Step Solution

Let's start by recalling how to define a root as a power:

an=a1n \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:

161=1612=16112=161=16=4 \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.

Answer:

4 4

Video Solution
Exercise #4

Solve the following exercise:

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

Step-by-Step Solution

Let's start by recalling how to define a square root as a power:

an=a1n \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:

12=122=1122=12=2 \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.

Answer:

2 \sqrt{2}

Video Solution
Exercise #5

Solve the following exercise:

301= \sqrt{30}\cdot\sqrt{1}=

Step-by-Step Solution

Let's start with a reminder of the definition of a root as a power:

an=a1n \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:

301=3012=30112=301=30 \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.

Answer:

30 \sqrt{30}

Video Solution

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