Rational exponents are exponents of numbers expressed as rational numbers, i.e., in a^p/q, a is the base and p/q is the rational exponent where q ≠ 0. In rational exponents, the base must be a positive integer.

In this article, we learn about Rational exponents and practice some questions.

What is a Rational Exponent?

An exponential expression of the form a^p has a rational exponent if p is a rational number. The powers and roots of a number are expressed together in rational exponents.

Rational Exponent

Examples of Rational Exponents are 8^2/3, 4^1/2,17^2/3, etc. There are a lot of examples we can create on our own.

To understand the terms base and exponent let us take an example: 8^2/3

where,

• 8 is the Base
• 2/3 is the Exponent

Important Formulas Related to Rational Exponents

Various formulas of Rational Exponents are:

• a^m × aⁿ = a^{(m + n)}
• a^m ÷ aⁿ = a^{(m – n)}
• a^m × b^m = (ab)^m
• a^m ÷ b^m= (a÷b)m
• a^-m = (1/a)^m
• a⁰ = 1
• (a^m)ⁿ = a^m

Rational Exponents Practice Problems: Solved

Here are worksheet on Rational Exponents, essential in algebra for simplifying expressions involving roots and powers. These exercises cover a range of problems designed to enhance understanding and proficiency in manipulating fractional exponents.

1. Evaluate 8^2/3.

Given: 8^2/3

= ((2)³)^2/3

= (2)^3×2/3 [using (a^m)^1/m=a^m×1/m]

= 2² = 4

2. Solve 16^1/2

Given: 16^1/2

= (4²)^1/2

= (4)^2×1/2 [using (a^m)^1/m=a^m×1/m]

= 4¹ = 4

3. Solve 36^1/2

Given: 36^1/2

= (6²)^1/2

= (6)^2×1/2 [using (a^m)^1/m=a^m×1/m]

= (6)¹ = 6

4. Solve 27^2/3 + 27⁰

Given: 27^2/3 + 27⁰ [a⁰ = 1 ]

= (9³)^2/3 + 27⁰

= (9)^3×2/3 + 1

= (9)² +1

5. Solve 64^1/3+49^1/2

Given: 64^1/3+ 49^1/2

= (4³)^1/3+ (7²)^1/2 [using (a^m)^1/m=a^m×1/m]

= 4¹ + 7¹

= 4+7 = 11

6. Solve 3^1/3 × 9^1/9 × 27^1/27

Given: 3^1/3 × 9^1/9 × 27^1/27

= 3^1/3 × (3²)^1/9 × (3³​)^1/27 [using (a^m)^1/m=a^m×1/m]

= 3^1/3 × 3^2/9 × 3^3/27

= 3^1/3 × 3^2/9 × 3^1/9

= 3^{(1/3 + 2/9 + 1/9)}

= 3^6/9 = 3^2/3

7. Solve a¹ + a^2×1/2 + a⁰

Given: a¹+ a^2×1/2 + a⁰ [a⁰ = 1]

= a + a¹ +1

= 2a + 1

8. Solve x^2×1/2 + y^3×1/3

Given: x^2×1/2+y^3×1/3

= x¹ + y¹

= x + y

9. Solve (9/4)^1/2

Given: (9/4)^1/2

= (3/2)^2×1/2 [using (a^m)^1/m=a^m×1/m]

= (3/2)¹

= 3/2