In this chapter, we delve into the concepts of exponents and roots to simplify expressions involving fractional exponents. Understanding these principles is crucial for solving complex mathematical problems efficiently.
Value of Expressions with Fractional Exponents
In this section, we will evaluate expressions involving fractional exponents. We'll simplify each expression step-by-step to find the exact values.
Here, we'll explore various techniques for simplifying expressions with exponents. The focus will be on applying exponent rules and arithmetic operations to achieve the desired simplifications.
Question 1. Find the Value of:
(i) 641/2
641/2
= (8 × 8)1/2
= (82)½
= 81
= 8
(ii) 321/5
321/5
= (2 x 2 x 2 x 2 x 2)1/5
= (25)⅕
= 21
= 2
(iii) 1251/3
(125)1/3
= (5 × 5 × 5)1/3
= (53)⅓
= 51
= 5
Question 2. Find the value of:
(i) 93/2
93/2
= (3 × 3)3/2
= (32)3/2
= 33
= 27
(ii) 322/5
322/5
= (2 × 2 × 2 × 2 × 2)2/5
= (25)2⁄5
= 22
= 4
(iii) 163/4
163/4
= (2 × 2 × 2 × 2)3/4
= (24)3⁄4
= 23
= 8
(iv) 125-1/3
125-1/3
= (5 × 5 × 5)-1/3
= (53)-1⁄3
= 5-1
= 1/5
Question 3. Simplify the followings:
(i) 22/3 × 21/5
22/3 × 21/5
= 2(2/3) + (1/5) ⸪ As, am × an = am + n
= 213/15
(ii) (1/33)7
(1/33)7
= (3-3)7 ⸪ As, (am)n = am x n
= 3-21
(iii) 111/2/111/4
111/2/111/4
= 11(1/2) - (1/4) ⸪ As, am × a-n = am - n
= 111/4
(iv) 71/2 × 81/2
71/2 × 81/2
= (7 × 8)1/2 ⸪ As, am × bm = (a × b)m
= 561/2
Summary
In this chapter, we explored the concepts of exponents and roots, focusing on simplifying expressions with fractional exponents. We learned various techniques for simplifying these expressions, including applying exponent rules and performing arithmetic operations. Mastering these principles is crucial for solving more complex mathematical problems efficiently and is essential for understanding further concepts in algebra and calculus.