Roots and Exponents

Question 1

If you put a non-negative number under the radical sign, can the output be negative? Y/N?

Answer 1

No.

Question 2

If you have a variable that’s squared(x^2), what is the output of the square root of that squared variable (sqrt(x^2))?

Answer 2

The size of the variable: |x|, which equals +x or -x.

Question 3

If you have only an integer under the radical sign, say 25: √25, what is the output?

Answer 3

5

Question 4

In the nth root of x^n, if n is even, what is certain about the output?

Answer 4

The output will always be positive and will be equal to: |x|

Question 5

In the nth root of x^n, if n is odd, what is certain about the output?

Answer 5

The output will take the sign of x.

Question 6

Recall the perfect squares up to 225.

Answer 6

0,1,4,9,16,25,36,49,64,81,100,121,144,169,196, and 225.

Question 7

What is a definite characteristic of the square root of a perfect square?

Answer 7

The square root of a perfect square is a whole number.

Recall: a perfect square is a number whose prime factorization contains only even powers.

Question 8

Recall the first 11 non-negative perfect cubes.

Answer 8

0,1,8,27,64,125,216,343,512,729, and 1000.

Question 9

How do you simplify radicals?

Answer 9

First locate and simplify any perfect squares or perfect cubes within the expression.

Question 10

Recall the square roots of single-digit number 2 to 8.

Answer 10

√2= 1.4, √3=1.7, √5=2.2, √6=2.4, √7=2.6, √8=2.8

Question 11

Recall the cube roots of single-digit numbers 2 to 9.

Answer 11

3√2 = 1.3, 3√3 = 1.4, 3√4 = 1.6, 3√5 = 1.7, 3√6 = 1.8, 3√7 = 1.9, 3√9 = 2.1

Question 12

Recall the fourth roots of single-digit numbers other than 1.

Answer 12

4√2 = 1.2, 4√3 = 1.3, 4√4 = 1.4, 4√5 = 1.5, 4√6 = 1.6, 4√8 = 1.7 and 4√9 = 1.7(also).

Question 13

What is the radical identity for multiplication?

Answer 13

m√a * m√b = m√ab

Question 14

What is the radical identity for division?

Answer 14

n√a / n√b = n√(a/b)

Question 15

What is the identity for combining radical and non-radicals in multiplication and division?

Answer 15

Multiply or divide radicals by radicals and non-radicals by non-radicals.

Multiplication: (an√b)(cn√d) = acn√(bd)
Division: (an√b)/(cn√d) = a/c*n√(b/d)

Question 16

If numbers under a square root are being added or subtracted, do we take the square root of each element or do we perform the addition/subtraction then take the square root?

Answer 16

Order of operations dictate taking the square root after performing the addition/subtraction operation.

Question 17

What’s the rule for rationalizing a denoninator?

Answer 17

x/√a.

If a is not a perfect square, to rationalize a one-term radical, √a, in the denominator of a fraction, multiply that fraction by √a/√a.

Question 18

Where binomials in the form: a + √b, a - √b, √a +b, √a - b, √a - √b, and √a + √b are in the denominator of a fraction, how do we simplify the fraction?

Answer 18

We simplify by using rationalization, i.e., by multiplying the numerator and the denominator of the fraction by the conjugate of the denominator.

Question 19

Can a simplified fraction ever have a radical in the denominator? For e.g. 1/√2. Y/N?

Answer 19

A simplified fraction can never have a radical in the denominator.

Question 20

We know that: √(x^2) = |x|, what is n√((x+y)^n) = ? (when n is even)

Answer 20

|x+y|

Question 21

/if the unknown is inside a square root in the equation, we have to square it to get rid of the square root since the square and the square root are inverse operations of each other, but there is one additional step to be done before selecting our answer. What is that step?

Answer 21

Plug the value of the unknown into the equation. If we have a quadratic, this is especially important since we must use one of the values of the unknown to make the equation hold.