R.6 Radicals and Rational Exponents

Question 1

What do we call an expression containing a radical?

Answer 1

A radical expression

Question 2

√(a^2) = ?

Answer 2

|a|

Question 3

what is the difference between √(A^2 + B^2) and √(A + B)^2?

Answer 3

The second would be the absolute value of (A+B). The second would be a number that squared is (A^2 + B^2)

Question 4

What determines if a √ can be positive or negative?

Answer 4

If the index is even, the root cannot be negative.

If the index is odd, the root can be negative.

Question 5

What is the n√a^n?

Answer 5

n√a^n = |a| when n is even

n√a^n = a when n is odd

Question 6

How can a radical expression be written using rational exponents?

Answer 6

√a^2 = (a^2)^(1/2)

If a is a real number, and n is an integer greater than 1, then:

n√a = a^(1/n)

Question 7

What is the power of (a^n)^(1/m)?

Answer 7

a^(n/m)

Question 8

How does the commutative property affect rational exponents?

Answer 8

The order in which the numerator and denominator (power and root) are applied do not matter.

Question 9

Is it generally easier to compute the root first, or the power first?

Answer 9

The root. It lets us determine if the number is real before proceeding.

Question 10

What is the product property of radicals?

Answer 10

When n is a rational number:

(xy)^(1/n) = x^(1/n) * y^(1/n)

and

x^(1/n) * y^(1/n) = (xy)^(1/n)

Note that this only applies to a product, not to a sum or difference

Question 11

What is the quotient property of radicals?

Answer 11

If n√A and n√B represent real-value expressions, with B != 0, then:

n√(A/B) = (n√A) / (n√B)

Question 12

What are like radicals?

Answer 12

Radicals that have the same index and radicand.

Question 13

What is rationalizing the denominator?

Answer 13

writing a radical expression in simplest form:

1) the radicand has no perfect nth root factors.
2) The radicand contains no fractions.
3) No radicals occur in the denominator.

Question 14

What is the Pythagorean Theorem?

Answer 14

For a right triangle, the squares of the two shorter sides equal the square of the hypotenuse:

A^2 + B^2 = C^2

Question 15

What are three methods of rationalizing the denominator?

Answer 15

If the denominator is a single term:

1) Multiply the numerator an denominator by the factors required to eliminate the radicals in the denominator.
2) If the radicand is a rational expression, build an equivalent fraction within the radical having perfect nth root factors in the denominator.
3) If the denominator is a sum or difference containing a rational term, and the radical is a square root, multiply by a conjugate. Remember that:

(A + B) (A - B) = (A^2 - B^2)

Question 16

What is Kepler’s third law of planetary motion?

Answer 16

T = 0.407R^(3 / 2)

T is the number of days required for a planet to make a full rotation of the sun, where R is the maximum radius of the planet’s orbit, in millions of miles.