<< Chapter < Page Chapter >> Page >

Understanding n Th roots

Suppose we know that a 3 = 8. We want to find what number raised to the 3rd power is equal to 8. Since 2 3 = 8 , we say that 2 is the cube root of 8.

The n th root of a is a number that, when raised to the n th power, gives a . For example, −3 is the 5th root of −243 because ( −3 ) 5 = −243. If a is a real number with at least one n th root, then the principal n th root    of a is the number with the same sign as a that, when raised to the n th power, equals a .

The principal n th root of a is written as a n , where n is a positive integer greater than or equal to 2. In the radical expression, n is called the index    of the radical.

Principal n Th root

If a is a real number with at least one n th root, then the principal n th root    of a , written as a n , is the number with the same sign as a that, when raised to the n th power, equals a . The index    of the radical is n .

Simplifying n Th roots

Simplify each of the following:

  1. −32 5
  2. 4 4 1 , 024 4
  3. 8 x 6 125 3
  4. 8 3 4 48 4
  1. −32 5 = −2 because ( −2 ) 5 = −32
  2. First, express the product as a single radical expression. 4,096 4 = 8 because 8 4 = 4,096
  3. 8 x 6 3 125 3 Write as quotient of two radical expressions . 2 x 2 5 Simplify .
  4. 8 3 4 2 3 4 Simplify to get equal radicands . 6 3 4   Add .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Simplify.

  1. −216 3
  2. 3 80 4 5 4
  3. 6 9 , 000 3 + 7 576 3
  1. −6
  2. 6
  3. 88 9 3
Got questions? Get instant answers now!

Using rational exponents

Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index n is even, then a cannot be negative.

a 1 n = a n

We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an n th root. The numerator tells us the power and the denominator tells us the root.

a m n = ( a n ) m = a m n

All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

Rational exponents

Rational exponents are another way to express principal n th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is

a m n = ( a n ) m = a m n

Given an expression with a rational exponent, write the expression as a radical.

  1. Determine the power by looking at the numerator of the exponent.
  2. Determine the root by looking at the denominator of the exponent.
  3. Using the base as the radicand, raise the radicand to the power and use the root as the index.

Writing rational exponents as radicals

Write 343 2 3 as a radical. Simplify.

The 2 tells us the power and the 3 tells us the root.

343 2 3 = ( 343 3 ) 2 = 343 2 3

We know that 343 3 = 7 because 7 3 = 343. Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.

343 2 3 = ( 343 3 ) 2 = 7 2 = 49

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write 9 5 2 as a radical. Simplify.

( 9 ) 5 = 3 5 = 243

Got questions? Get instant answers now!

Writing radicals as rational exponents

Write 4 a 2 7 using a rational exponent.

The power is 2 and the root is 7, so the rational exponent will be 2 7 . We get 4 a 2 7 . Using properties of exponents, we get 4 a 2 7 = 4 a −2 7 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write x ( 5 y ) 9 using a rational exponent.

x ( 5 y ) 9 2

Got questions? Get instant answers now!

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask