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
Examples and Exercises with answers.

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

Practice makes one perfect.

This is especially true for remembering
Algebra Formulas (Math Formulas).

So, take the exercises seriously
and practice solving the problems.

Set of Solved Examples : Fractional Exponents

Solved Example 1 of Fractional Exponents

If a^x = b^y = c^z and b⁄a = c⁄b, show that y⁄x = 2z⁄(x + z)

Solution to Example 1 of Fractional Exponents:

Let a^x = b^y = c^z = k
⇒ a = k^1⁄x; b = k^1⁄y; c = k^1⁄z;
By data, b⁄a = c⁄b ⇒ (k^1⁄y)⁄(k^1⁄x) = (k^1⁄z)⁄(k^1⁄y)
We know a^m⁄aⁿ = a^{m - n}
Applying this here, we get
(k^{1⁄y - 1⁄x}) = (k^{1⁄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
x⁄y = (x + z)⁄2z ⇒ y⁄x = 2z ⁄(x + z) (Proved.)

Solved Example 2 of Fractional Exponents

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

Solution to Example 2 of Fractional Exponents:

Let a = 3^1⁄3. Then a³ = 3........(i)
and b = 1⁄(3^1⁄3). Then b³ = 1⁄3 .......(ii)
Also, ab = 1.......(iii) and a + b = y........(iv).
y = 3^1⁄3 + 1⁄(3^1⁄3) = 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 2x^1⁄3 + 2x^-1⁄3 = 5.

Solution to Example 3 of Fractional Exponents:

Let a = x^1⁄3. 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.
This is a quadratic equation.
We know

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

Exercise on Fractional Exponents

1. Show that
   1⁄(1 + x^{a - b} + x^{a - c}) + 1⁄(1 + x^{b - c} + x^{b - a}) + 1⁄(1 + x^{c - a} + x^{c - b}) = 1.
2. If a^x = b^y = c^z = d^w and ab = cd,
   show that: 1⁄x + 1⁄y = 1⁄w + 1⁄z .
   
3. If a^1⁄3 + b^1⁄3 + c^1⁄3 = 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.