Rational Exponent
In rational exponent there are positive rational exponent and negative rational exponent.
Positive Rational Exponent:
We know that 2³ = 8. It can also be expressed as 8\(^{\frac{1}{3}}\) = 2.
In general if x and yare non-zero rational numbers and m is a positive integer such that xᵐ = y then we can also express it as y\(^{\frac{1}{m}}\) = x but we can write y\(^{\frac{1}{m}}\) = \(\sqrt[m]{y}\) and is called mᵗʰ root of y.
For example, y\(^{\frac{1}{2}}\) = \(\sqrt[2]{y}\), y\(^{\frac{1}{3}}\) = ∛y, y\(^{\frac{1}{4}}\) = ∜y, etc. If x is a positive rational number then for a positive ration exponent p/q we have x₀can be defined in two equivalent form.
x\(^{\frac{p}{q}}\) = \((x^{p})^{\frac{1}{q}}\) = \(\sqrt[q]{x^{p}}\) is read as qᵗʰ root of xᵖ
x\(^{\frac{p}{q}}\) = \((x^{\frac{1}{q}})^{p}\) = \((\sqrt[q]{x})^{p}\) is read as pᵗʰ power of qᵗʰ root of x
For example:
1. Find (125)\(^{\frac{2}{3}}\)
Solution:
(125)\(^{\frac{2}{3}}\)
125 can be expressed as 5 × 5 × 5 = 5³
2. Find (8/27)4/3
Solution:
(8/27)4/3
8 = 23 and 27 = 33
So, we have (8/27)4/3 = (23/33)4/3
= [(2/3) 3]4/3
= (2/3)
= (2/3) 4
= 2/3 × 2/3 × 2/3 × 2/3
= 16/81
3. Find 91/2
Solution:
91/2
= √(2&9)
= [(3)2]1/2
= (3)
= 3
4. Find 1251/3
Solution:
1251/3
= ∛125
= [(5) 3]1/3
= (5)
= 5
Negative Rational Exponent:
We already learnt that if x is a non-zero rational number and m is any positive integer then x-m = 1/xm = (1/x)m, i.e., x-m is the reciprocal of xm.Same rule exists of rational exponents.
If p/q is a positive rational number and x > 0 is a rational number
Then x-p/q = 1/xp/q = (1/x) p/q, i.e., x-p/q is the reciprocal of xp/q
If x = a/b, then (a/b)-p/q = (b/a)p/q
For example:
1. Find 9-1/2
Solution:
9-1/2
= 1/91/2
= (1/9)1/2
= [(1/3)2]1/2
= (1/3)
= 1/3
2. Find (27/125)-4/3
Solution:
(27/125)-4/3
= (125/27)4/3
= (53/33)4/3
= [(5/3) 3]4/3
= (5/3)
= (5/3)4
= (5 × 5 × 5 × 5)/(3 × 3 × 3 × 3)
= 625/81
● Exponents
Integral Exponents of a Rational Numbers
● Exponents - Worksheets
8th Grade Math Practice
From Rational Exponent to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Recent Articles
-
Half Past an Hour | What does Half Past Mean? | Half an Hour|Half Past
Nov 25, 24 01:38 PM
We learnt that, one hour is equal to 60 minutes. When one hour is divided into two, it is half an hour or 30 minutes. The minute hand points at 6. We say, 30 minutes past an hour or half past an hour… -
Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr
Nov 25, 24 01:09 PM
There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t… -
Worksheet on Yesterday, Today and Tomorrow | Sequence of the Week-days
Nov 25, 24 01:18 AM
Practice the worksheet on yesterday, today and tomorrow, the questions are based on the sequence of the week-days, their names and order. We know, the present day is today, the day before today -
Yesterday, Today and Tomorrow | Name of each Day of the Week
Nov 25, 24 01:09 AM
We will learn the relation between yesterday, today and tomorrow. There are seven days in a week. We know the name of each day of the week. Sunday is known as the first day of the week. -
The Story about Seasons | Spring | Summer | Autumn | Winter
Nov 25, 24 12:48 AM
Kids let’s enjoy the story about seasons. Here we will discuss about the four seasons and the duration. Some months are too hot and some are too cold. The period of hot months is called the hot
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.