Cogito
Integrated Math 3 · Chapter 3 · Lesson 1
Rational Exponents and Radicals
A fraction in the exponent is a root.
12 problems · about 21 minutes · N-RN.A.1, N-RN.A.2
What this lesson teaches
The student converts between radical and rational exponent form and simplifies both.
- x^(m/n) is the nth root of x, raised to the m.
- Taking the root first keeps the numbers manageable.
- Even roots restrict the domain; odd roots do not.
Warm Up
Straightforward practice. Get the method working first.
5 problemsWhat is 27^(2/3)?
Answer 9
Why 9.
In x^(m/n), what does the denominator give?
Answer The root to take.
Why The root.
What is 25^(1/2)?
Answer 5
Why The square root.
What is 27^(1/3)?
Answer 3
Why The cube root.
What is 4^(3/2)?
Answer 8
Why Root then cube.
Build It Up
The same ideas with more to keep track of.
3 problemsWhat is 9^(3/2)?
Answer 27
Why 3 cubed.
What is 4^(−1/2), as a decimal?
Answer 0.5
Why 1 over the square root.
Does an odd root accept negative inputs? 1 yes, 0 no.
Answer 1
Why Cubing keeps the sign.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each expression with its value.
Answer 8 to the one third → Two; 8 to the two thirds → Four; 8 to the zero → One
Why The denominator is the root and the numerator is the power.
What is 32^(1/5)?
Answer 2
Why 2 to the fifth is 32.
The Power: What is 8^(2/3)?
Answer 4
Why 4.
The Root: What is 16^(1/2)?
Answer 4
Why 4.