Co n cep t Byt e
Use W ith Lesson 7 -5Rel at i n g Rad i cal s t o
Rat i o n al Ex p o n en t s
@ Common Core State Standards
Pr ep ar es f o r N- RN.A.2 Rewrite expressions
in volving radicals and rational exponents using
the properties o f exponents.You can use patterns to explore the relationship between radicals and rational
exponents.Activity 1
Copy and complete the table. Round to the nearest thousandth if necessary.
9 V9 = gx =
36 \/36 = 36^ =
(^15) \ 15 = 8 : 157 „
(^24) V 24 » I 242 -^1
A calculat or can help wit h
fract ional powers. On most
calculat ors, the # key
is used to enter an exponent.
For inst ance, 2 7 3 would be
entered as 27 9 (1 3).
Look f o r a Pat t ern What appears to be the relationship between the square root
of a number and the | power of the number?
a. Make a Conject ure What do you think the relationship is between the cube
root of a number and the | power of the number?
b. Test your conjecture for the numbers 8 and 27, which have integer cube roots.
Describe your results.
You can also find a pattern when placing powers with rational exponents on a number line.Activity 2
For Exercises 7-10, use a calculator to find the approximate value of each
expression. Then plot each number on a number line.- 32, 33, 3°, 31, 35 8. 5?, 52, 5°, 55, 5?
- 255, 255, 252, 25, 255 10. 10, 105, 10*, 105, 1Q5
- Look fo r a Pat t ern For a set of powers with the same base that is greater than 1,
how can you predict their order on a number line?
MP 7Ex e r c i se s
Predict answers for the following problems using the patterns you have discovered.- If a > 1 and x < y, which is greater, ax or
- Ifl < x < y and a > 1 , which is greater, xa or yal
- If a > 1 and x > 1 , which is greater, ax or a x?
ACo n cep t By t e Re l a t i n g Rad i cal s t o Rat i o n al Exp o n en t s 447