# FRACTIONAL EXPONENTS - SOLVED EXAMPLES AND EXERCISES ON RATIONAL EXPONENTS

Please study  the Basics of Fractional Exponents,
if you have not already done so.

It is a prerequisite here.

There, we provided the explanation for Rational Exponents.

We applied the same 7 Laws and the 2 Rules
in solving problems for Rational Exponents.

We provided a few solved examples
and problems for practice with answers.

Here we provide many more Solved

Studying the worked out problems will help remember and
apply the 7 Laws and the 2 Rules for Rational Exponents.

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## Set of Solved Examples : Fractional Exponents

### Solved Example 1 of Fractional Exponents

If ax = by = cz and ba = cb, show that yx = 2z⁄(x + z)

Solution to Example 1 of Fractional Exponents:

Let ax = by = cz = k
a = k1⁄x; b = k1⁄y; c = k1⁄z;
By data, ba = cb ⇒ (k1⁄y)⁄(k1⁄x) = (k1⁄z)⁄(k1⁄y)
We know aman = am - n
Applying this here, we get
(k1⁄y - 1⁄x) = (k1⁄z - 1⁄y)
Since the bases are same, the exponents have to be equal.
∴ (1⁄y - 1⁄x) = (1⁄z - 1⁄y)
⇒ 1⁄y + 1⁄y = 1⁄z + 1⁄x ⇒ 2⁄y = (x + z)⁄xz
Multiplying both sides with x⁄2, we get
xy = (x + z)⁄2zyx = 2z ⁄(x + z) (Proved.)

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### Solved Example 2 of Fractional Exponents

If y = 31⁄3 + 1⁄(31⁄3), show that 3y3 - 9y = 10.

Solution to Example 2 of Fractional Exponents:

Let a = 31⁄3. Then a3 = 3........(i)
and b = 1⁄(31⁄3). Then b3 = 1⁄3 .......(ii)
Also, ab = 1.......(iii) and a + b = y........(iv).
y = 31⁄3 + 1⁄(31⁄3) = a + b
y3 = (a + b)3 = a3 + b3 + 3ab(a + b)
Using (i), (ii), (iii) and (iv) here, we get
y3 = 3 + 1⁄3 + 3(1)(y)
Multiplying both sides with 3, we get
3y3 = 9 + 1 + 9y ⇒ 3y3 - 9y = 10. (Proved.)

### Solved Example 3 of Fractional Exponents

Solve 2x1⁄3 + 2x-1⁄3 = 5.

Solution to Example 3 of Fractional Exponents:

Let a = x1⁄3. Then x-1⁄3 = 1⁄a.
The given equation becomes 2a + 2⁄a = 5.
Multiplying both sides with a, we get
2a2 + 2 = 5a. ⇒ 2a2 - 5a + 2 = 0.
We know

The solution of the quadratic equation ax2 + bx + c = 0 is
x = {-b ± √( b2 - 4ac) }⁄2a

Applying this here, we get
a = [-(-5) ± √{(-5)2 - 4(2)(2)}]⁄[2(2)]
= [5 ± √{ 25 - 16 }]⁄[4] = [5 ± √{ 9 }]⁄[4] = [5 ± 3]⁄[4]
= (5 + 3)⁄4 or (5 - 3)⁄4 = 8⁄4 or 2⁄4 = 2 or 1⁄2 .
a = x1⁄3a3 = x;
x = 23 or (1⁄2)3 = 8 or 1⁄8. Ans.

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## Exercise on Fractional Exponents

1. Show that
1⁄(1 + xa - b + xa - c) + 1⁄(1 + xb - c + xb - a) + 1⁄(1 + xc - a + xc - b) = 1.
2. If ax = by = cz = dw and ab = cd,
show that: 1⁄x + 1⁄y = 1⁄w + 1⁄z .
3. If a1⁄3 + b1⁄3 + c1⁄3 = 0, show that: (a + b + c)3 = 27abc.

For Answers, see at the bottom of the page.

### Answers to Exercise on Fractional Exponents

1. (i) 1. (ii) 1.
2. (i) 1⁄2. (ii) -8.

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