Commented and Resolved Radiciation Exercises

THE radiciation is the operation we use to find a number that multiplied by itself a certain number of times, equals a known value.

Take advantage of the solved and commented exercises to answer your doubts about this mathematical operation.

question 1

Factor the root of and find the root result.

See Answer

Correct answer: 12.

1st step: factor the number 144

2nd step: write 144 in power form

Note that 2⁴ can be written as 2².2², because 2²⁺²= 2⁴

Therefore,

3rd step: replace radicand 144 by the power found

In this case we have a square root, that is, a root of index 2. Therefore, as one of the properties of radiciation is we can eliminate the root and solve the operation.

question 2

What is the value of x on equality ?

a) 4
b) 6
c) 8
d) 12

See Answer

Correct answer: c) 8.

Observing the exponent of the radicands, 8 and 4, we can see that 4 is half of 8. Therefore, the number 2 is the common divisor between them and this is useful to find out the value of x, because according to one of the properties of the radiciation .

Dividing the index of the radical (16) and the exponent of the radicand (8), we find the value of x as follows:

Therefore, x = 16: 2 = 8.

question 3

simplify the radical .

See Answer

Right answer: .

To simplify the expression, we can remove from the root the factors that have an exponent equal to the index of the radical.

For that, we must rewrite the radicand so that the number 2 appears in the expression, since we have a square root.

Replacing the previous values ​​in the root, we have:

Like , we simplify the expression.

question 4

Knowing that all expressions are defined in the set of real numbers, determine the result to:

The)

B)

ç)

d)

See Answer

Right answer:

The) can be written as

Knowing that 8 = 2.2.2 = 2³ we replaced the value of 8 in the root with the power 2³.

B)

ç)

d)

question 5

rewrite the radicals ; and so that all three have the same index.

See Answer

Right answer: .

To rewrite the radicals with the same index, we need to find the least common multiple between them.

MMC = 2.2.3 = 12

Therefore, the index of the radicals must be 12.

However, to modify the radicals we need to follow the property .

To change the radical index we must use p = 6, since 6. 2 = 12

To change the radical index we must use p = 4, since 4. 3 = 12

To change the radical index we must use p = 3, since 3. 4 = 12

question 6

What is the result of the expression ?

The)
B)
ç)
d)

See Answer

Correct answer: d) .

For the property of the radicals , we can solve the expression as follows:

question 7

Rationalize the expression's denominator .

See Answer

Right answer: .

To remove the radical from the quotient denominator, we must multiply the two terms of the fraction by a rationalizing factor, which is calculated by subtracting the index of the radical by the exponent of the radicand: .

Therefore, to rationalize the denominator the first step is to calculate the factor.

Now, we multiply the quotient terms by the factor and solve the expression.

Therefore, rationalizing the expression we have as a result .

Commented and resolved university entrance exam questions

question 8

(IFSC - 2018) Review the following statements:

I.

II.

III. effecting itself , you get a multiple of 2.

Check the CORRECT alternative.

a) All are true.
b) Only I and III are true.
c) All are false.
d) Only one of the statements is true.
e) Only II and III are true.

See Answer

Correct alternative: b) Only I and III are true.

Let's solve each of the expressions to see which ones are true.

I. We have a numeric expression involving several operations. In this type of expression, it is important to remember that there is a priority to perform the calculations.

So we must start with rooting and potentiation, then multiplication and division, and finally addition and subtraction.

Another important observation is regarding - 5². If there were parentheses, the result would be +25, but without the parentheses, the minus sign is the expression and not the number.

So the statement is true.

II. To solve this expression, we will consider the same remarks made in the previous item, adding that we solve the operations inside the parentheses first.

In this case, the statement is false.

III. We can solve the expression using the distributive property of multiplication or the remarkable product of the sum by the difference of two terms.

So we have:

Since the number 4 is a multiple of 2, this statement is also true.

question 9

(CEFET/MG - 2018) If , then the value of the expression x² + 2xy +y² – z² é

The)
B)
c) 3
d) 0

See Answer

Correct alternative: c) 3.

Let's start the question by simplifying the root of the first equation. For this, we will pass the 9 to the power form and we will divide the index and the root root by 2:

Considering the equations, we have:

Since the two expressions, before the equal sign, are equal, we conclude that:

Solving this equation, we'll find the value of z:

Replacing this value in the first equation:

Before replacing these values ​​in the proposed expression, let's simplify it. Note that:

x² + 2xy + y² = (x + y)²

So we have:

question 10

(Sailor's Apprentice - 2018) If , so the value of A² é:

to 1
b) 2
c) 6
d) 36

See Answer

Correct alternative: b) 2

As the operation between the two roots is multiplication, we can write the expression in a single radical, that is:

Now, let's square A:

Since the index of the root is 2 (square root) and it is squared, we can take the root. Thus:

To multiply, we will use the distributive property of multiplication:

question 11

(Apprentice Sailor - 2017) Knowing that the fraction is proportional to the fraction , it is correct to say that y is equal to:

a) 1 - 2
b) 6 + 3
c) 2 -
d) 4 + 3
e) 3 +

See Answer

Correct alternative: e)

As fractions are proportional, we have the following equality:

Passing the 4 to the other side and multiplying, we find:

Simplifying all terms by 2, we have:

Now, let's rationalize the denominator, multiplying up and down by the conjugate of :

question 12

(CEFET/RJ - 2015) Let m be the arithmetic mean of numbers 1, 2, 3, 4 and 5. Which option comes closest to the result of the expression below?

a) 1.1
b) 1.2
c) 1.3
d) 1.4

See Answer

Correct alternative: d) 1.4

To start, we will calculate the arithmetic mean between the indicated numbers:

Replacing this value and solving the operations, we find:

question 13

(IFCE - 2017) Approximating the values ​​of to the second decimal place, we get 2.23 and 1.73, respectively. Approaching the value of to the second decimal place, we get

a) 1.98.
b) 0.96.
c) 3.96.
d) 0.48.
e) 0.25.

See Answer

Correct alternative: e) 0.25

To find the expression value, we will rationalize the denominator, multiplying by the conjugate. Thus:

Solving the multiplication:

Replacing the root values ​​by the values ​​informed in the problem statement, we have:

question 14

(CEFET/RJ - 2014) By which number should we multiply the number 0.75 so that the square root of the product obtained is equal to 45?

a) 2700
b) 2800
c) 2900
d) 3000

See Answer

Correct alternative: a) 2700

First, let's write 0.75 as an irreducible fraction:

We'll call the number we're looking for x and write the following equation:

By squaring both members of the equation, we have:

question 15

(EPCAR - 2015) The sum value is a number

a) natural less than 10
b) natural greater than 10
c) non-integer rational
d) irrational.

See Answer

Correct alternative: b) natural greater than 10.

Let's start by rationalizing each portion of the sum. For this, we will multiply the numerator and denominator of the fractions by the conjugate of the denominator, as indicated below:

To effect the multiplication of the denominators, we can apply the remarkable product of the sum by the difference of two terms.

S = 2 - 1 + 14 = 15

You may also be interested in:

• Potentiation Exercises
• Potentiation Properties
• Simplification of Radicals
• Exercises on Simplification of Radicals