Exercises on radical simplification

Correct answer: c) 3 square root of 3 .

When we factor a number we can rewrite it in power form according to the repeating factors. For 27, we have:

table row with 27 row with 9 row with 3 row with 1 end of table in right frame closes frame table row with 3 row with 3 row with 3 row with blank end of table

Therefore, 27 = 3.3.3 = 3³

This result can still be written as a multiplication of powers: 3².3, since 3¹=3.

Therefore, square root of 27 can be written as square root of 3 squared.3 end of root

Note that inside the root there is a term with an exponent equal to the index of the radical (2). In this way, we can simplify by removing the base of this exponent from within the root.

We arrived at the answer to this question: the simplified form of square root of 27 é 3 square root of 3 .

Correct answer: b) $numerator 4 square root of 2 over denominator 3 square root of 3 end of fraction$ .

According to the property presented in the question statement, we have to $square root of 32 over 27 end of root equal to numerator square root of 32 over denominator square root of 27 end of fraction$ .

To simplify this fraction, the first step is to factor out the radicands 32 and 27.

table row with 32 row with 16 row with 8 row with 4 row with 2 row with 1 end of the table in a frame right closes frame table row with 2 row with 2 row with 2 row with 2 row with 2 row with blank end of table

According to the factors found, we can rewrite the numbers using powers.

32 space equals space 2.2.2.2.2 space space 32 space equals space 2 to the power of 5 space equals space 2 squared.2 squared.2

27 space equal to space 3.3.3 space space 27 space equal to space 3 squared space equal to space 3 squared.

Therefore, the given fraction corresponds to $square root numerator of 32 over square root denominator of 27 end of fraction equals square root numerator of 2 squared.2 squared.2 end of root over denominator square root of 3 squared.3 end of root end of fraction$

We see that within the roots there are terms with an exponent equal to the index of the radical (2). In this way, we can simplify by removing the base of this exponent from within the root.

$numerator 2.2 square root of 2 over denominator 3 square root of 3 end of fraction$

We arrived at the answer to this question: the simplified form of square root of 32 over 27 end of root é $numerator 4 square root of 2 over denominator 3 square root of 3 end of fraction$ .

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Correct answer: b) square root of 8

We can add an external factor inside the root as long as the exponent of the added factor is equal to the index of the radical.

straight x straight space n nth root of straight y space equal to straight space n nth root of straight y space. straight space x to the power of straight n end of root

Replacing the terms and solving the equation, we have:

2 square space root of 2 space equal to square space root of 2 space. space 2 squared end of root space equals square space root of 2. space 4 end of root space equal to square space root of 8 space

Check out another way to interpret and resolve this issue:

The number 8 can be written in the form of the power 2³, because 2 x 2 x 2 = 8

Replacing the radicand 8 with the power 2³, we have square root from 2 to cube end of root .

Power 2³, can be rewritten as a multiplication of equal bases 2². 2 and if so, the radical will be square root from 2 squared.2 end of root .

Note that the exponent is equal to the index (2) of the radical. When this happens we must remove the base from inside the radicand.

Therefore 2 square root of 2 is the simplified form of square root of 8 .

Correct answer: c) 3 cubic space root of 4 .

Factoring the root 108, we have:

table row with 108 row with 54 row with 27 row with 9 row with 3 row with 1 end of the table in a frame right closes frame table row with 2 row with 2 row with 3 row with 3 row with 3 row with blank end of table

Therefore, 108 = 2. 2. 3. 3. 3 = 2².3³ and the radical can be written as cubic root of 2 squared.3 cubed end of root .

Note that in the root we have an exponent equal to the index (3) of the radical. Therefore, we can remove the base of this exponent from within the root.

3 radical index space 3 of 2 squared end of root

Power 2² corresponds to the number 4, so the correct answer is 3 cubic space root of 4 .

Correct answer: d) 2 square root of 6 .

According to the statement square root of 12 is the double of square root of 3 , therefore square root of 12 space equal to space 2 square root of 3 .

To find out which result when multiplied twice corresponds to square root of 24 , we must first factor the radicand.

table row with 24 row with 12 row with 6 row with 3 row with 1 end of table in right frame closes frame table row with 2 row with 2 row with 2 row with 3 row with blank end of table

Therefore, 24 = 2.2.2.3 = 2³.3, which can also be written as 2².2.3 and therefore the radical is square root of 2 squared.2.3 end of root .

In the radicand we have an exponent equal to the index (2) of the radical. Therefore, we can remove the base of this exponent from within the root.

By multiplying the numbers within the root, we arrive at the correct answer, which is 2 square root of 6 .

Correct answer: a) 3 square root of 5 comma space 4 square root of 5 straight space and space 6 square root of 5

First, we must factor out the numbers 45, 80 and 180.

table row with 45 row with 15 row with 5 row with 1 end of table in right frame closes frame table row with 3 row with 3 row with 5 row with blank end of table

line table with 80 line with 40 line with 20 line with 10 line with 5 line with 1 end of the table in a frame right closes frame table row with 2 row with 2 row with 2 row with 2 row with 5 row with blank end of table

line table with 180 line with 90 line with 45 line with 15 line with 5 line with 1 end of the table in a frame right closes frame table row with 2 row with 2 row with 3 row with 3 row with 5 row with blank end of table

According to the factors found, we can rewrite the numbers using powers.

45 = 3.3.5

45 = 3². 5

80 = 2.2.2.2.5

80 = 2². 2². 5

180 = 2.2.3.3.5

180 = 2². 3². 5

The radicals presented in the statement are:

square root of 45 space equal to square root space of 3 squared.5 end of root

square root of 80 space equal to square root space of 2 squared.2 squared.5 end of root

square root of 180 space equal to square root space of 2 squared.3 squared.5 end of root

We see that within the roots there are terms with an exponent equal to the index of the radical (2). In this way, we can simplify by removing the base of this exponent from within the root.

square root of 45 space equals space 3 square root of 5

square root of 80 space equals space 2.2 square root of 5 space equals space 4 square root of 5

square root of 180 space equals space 2.3 square root of 5 space equals space 6 square root of 5

Therefore, 5 is the root common to the three radicals after performing the simplification.

Correct answer: d) 16 square root of 6 .

First, let's factor out the measurement values in the figure.

table line with 54 line with 27 line with 9 line with 3 line with 1 end of table in right frame closes frame table line with 2 line with 3 line with 3 line with 3 line with blank end of table

table row with 150 row with 75 row with 25 row with 5 row with 1 end of the table in frame right closes frame table row with 2 row with 3 row with 5 row with 5 row with blank end of table

According to the factors found, we can rewrite the numbers using powers.

We see that within the roots there are terms with an exponent equal to the index of the radical (2). In this way, we can simplify by removing the base of this exponent from within the root.