For each of these statements, indicate whether T or F

[georgepa]

NP Strategy Guide 4th edition In Action Roots Problem Set Chapter 6 Page 83 Question 1


For each of these statements, indicate whether the statement is True or False:

(a) If x^2 = 11, then X = √11

On page 77 it says: Rule: Even roots only have a positive value. √4 = 2 NOT +/- 2

The solution states:

Even exponents hide the sign of the original number, because they always result in a positive value. If x^2 = 11, then |x| = √11 so x = +/- √11

I remember this in class so I chose only the +ve value. Am I understanding this wrong. Is it the case when only even root of x (e.g √x) is asked then the answer is only +ve. On the other hand if the value is asked given an even power of x then the answer is +/-?

[ankitmisri]

Hi,

For each of these statements, indicate whether the statement is True or False:

(a) If x^2 = 11, then X = √11

On page 77 it says: Rule: Even roots only have a positive value. √4 = 2 NOT +/- 2

The above rule applies only to numbers not variables. hence √(x^2) will always of 2 solutions....simple way to remember x^2 is quadratic so will always have 2 solutions

The solution states:

Even exponents hide the sign of the original number, because they always result in a positive value. If x^2 = 11, then |x| = √11 so x = +/- √11

The above is also correct since for both x= 2 and x= -2, x^2 = 4 thereby hiding sign, but x^3 = 8 (for x= 2) and x^3 = -8 (for x= -2).
So when calculating the nth root of any number in an expression (x^n = a)
if n is even please remember to factor in the sign for the solution
if n is odd retain the sign of "a" in the solution

[georgepa]

Hi,

For each of these statements, indicate whether the statement is True or False:

(a) If x^2 = 11, then X = √11

On page 77 it says: Rule: Even roots only have a positive value. √4 = 2 NOT +/- 2

The above rule applies only to numbers not variables. hence √(x^2) will always of 2 solutions....simple way to remember x^2 is quadratic so will always have 2 solutions

if n is odd retain the sign of "a" in the solution
[/i]

Can't 4 be written as x^2 ? Then, √(2^2) = +/- 2

I guess I'm asking why is -2 not an answer to √4

[ankitmisri]

Hi Please see the below

Every non-negative real number x has a unique non-negative square root, called the principal square root, which is denoted with a radical symbol as √, or, using exponent notation, as x^1/2. For example, the principal square root of 9 is 3, denoted √9=3, because 32 = 3 × 3 = 9 and 3 is non-negative. The principal square root of a positive number, however, is only one of its two square roots.

Regards,
Ankit

[georgepa]

Hi Please see the below

Every non-negative real number x has a unique non-negative square root, called the principal square root, which is denoted with a radical symbol as √, or, using exponent notation, as x^1/2. For example, the principal square root of 9 is 3, denoted √9=3, because 32 = 3 × 3 = 9 and 3 is non-negative. The principal square root of a positive number, however, is only one of its two square roots.

Regards,
Ankit

OK - so then the answer is +/- 2 and not +2 alone? But the guide says different. There is no mention why we discount the -2

On page 77 it says: Rule: Even roots only have a positive value. √4 = 2 NOT +/- 2

[Ben Ku]

The issue of positives and negatives when related to squares and roots is one that confuses a lot of people. Instead of remembering the rules, it's helpful to understand the concepts.

(1) When we take a number and square it, then we "lose" its sign; we don't know whether the original number was positive or negative.

3^2 = 9
(-3)^2 = 9

So if x^2 = 9, then what is x? The answer is we don't know. x could either be 3 or -3, because the square of both numbers is 9.

Mathematically, if x^2 = 9, we need to take the square root of both sides. The rule is, when you take the square root of BOTH sides of an equation, you must include both the positive and negative answers.
x^2 = 9
sqrt (x^2) = +/- sqrt(9)
x = 3 or -3

(2) Whenever we have the square root of a number, the number must be positive. We cannot take the square root of a negative number. If sqrt(x) = 6, then x must 36. x cannot be -36, because we cannot take the square root of a negative number.

Mathematically, if sqrt(x) = 6, we need to SQUARE both sides. When we take the square of both sides, they will be positive, since you lose the sign when you square a number.
sqrt(x) = 6
(sqrt(x))^2 = (6)^2
x = 36

Hope that helps.

henever we have an expression:
x^2 = 9

We can


NP Strategy Guide 4th edition In Action Roots Problem Set Chapter 6 Page 83 Question 1


For each of these statements, indicate whether the statement is True or False:

(a) If x^2 = 11, then X = √11

On page 77 it says: Rule: Even roots only have a positive value. √4 = 2 NOT +/- 2

The solution states:

Even exponents hide the sign of the original number, because they always result in a positive value. If x^2 = 11, then |x| = √11 so x = +/- √11

I remember this in class so I chose only the +ve value. Am I understanding this wrong. Is it the case when only even root of x (e.g √x) is asked then the answer is only +ve. On the other hand if the value is asked given an even power of x then the answer is +/-?