14. Let's evaluate.
a) 4^(1/2)
b) 25^(3/2)
c) (4/9)^(1/2)
d) (8/27)^(1/3)
e) (16/81)^(1/4)
f) (32/243)^(1/5)

15. Let's simplify.
a) (a^5 × a^7) / a^9
b) (p^6 × p^4) / p^3
c) (x^7 × x^(-2)) / x^3
d) (y^10 × y^(-4)) / y^4
e) (4x^5 × 3x) / 6x^2

16. Let's simplify.
a) (2^2)^3
b) (3^4)^3
c) (5 / 5^4)^2
d) (9^2 / 3^2)^3
e) (4^3 / 2^3)^4
f) (8^2 / 2^9)^5
g) (27^2 / 3^10)

17. Let's simplify.
a) (2^3 × 4^3 × 8^4) / (2^2 × 4^5 × 16^2)
b) (3^2 × 9^3 × 27^4) / (3^3 × 9^4 × 81^2)
c) (2^2 × 3^4 × 12^3) / (2^5 × 6^3)
d) (6^4 × 9^2 × 25^3) / (3^2 × 4^2 × 15^6)
e) (4^3 / 25^2)^2 × (3^4 / 3^2)^2 × (9^2 / 27^2)^2
f) (5^3 / 25^2)^2 × (8^2 / 4^2)^3 × (5 / 2)^4

18. Let's simplify.
a) x^(a-b) × x^(b-a)
b) x^(a-b) × x^(b-c) × x^(c-a)
c) (x^a)^(b-c) × (x^b)^(c-a) × (x^c)^(a-b)
d) (x^(p-q))^r × (x^(q-r))^p × (x^(r-p))^q
e) (x^(a+b) × x^(b+c) × x^(c+a)) / (x^(2a) × x^(2b) × x^(2c))
f) ((x^2)^(a+b) × (x^2)^(b+c) × (x^2)^(c+a)) / (x^a × x^b × x^c)^4

19. a) If a = 1 and b = 2, find the value of:
(i) a^b
(ii) b^a
(iii) (a+b)^(a+b)

b) If x = 5 and y = 3, find the value of:
(i) x^y
(ii) y^x
(iii) (x-y)^(x-y)

Let's go through these problems step-by-step.
Problem 14: Evaluate the Expressions
a) 4 1/2 The expression 4 1/2 means the square root of 4. 4 ​ = 2
b) 2 5 3/2 First, find the square root of 25, which is 5, and then raise it to the power 3. ( 25 ​ ) 3 = 5 3 = 125
c) ( 4/9 ) 1/2 This expression represents the square root of 9 4 ​ . Calculate separately: 9 ​ 4 ​ ​ = 3 2 ​
d) ( 8/27 ) 1/3 This means the cube root of 27 8 ​ . Calculate separately: 3 27 ​ 3 8 ​ ​ = 3 2 ​
e) ( 16/81 ) 1/4 This represents the fourth root of 81 16 ​ . Calculate separately: 4 81 ​ 4 16 ​ ​ = 3 2 ​
f) ( 32/243 ) 1/5 This means the fifth root of 243 32 ​ . Calculate separately: 5 243 ​ 5 32 ​ ​ = 3 2 ​
Problem 15: Simplify the Expressions
a) ( a 5 × a 7 ) / a 9 Use the property a m × a n = a m + n .
a 5 + 7 − 9 = a 3
b) ( p 6 × p 4 ) / p 3 p 6 + 4 − 3 = p 7
c) ( x 7 × x − 2 ) / x 3 x 7 − 2 − 3 = x 2
d) ( y 10 × y − 4 ) / y 4 y 10 − 4 − 4 = y 2
e) ( 4 x 5 × 3 x ) /6 x 2 First, simplify the coefficients: 6 4 × 3 ​ = 2 Combine the exponents: x 5 + 1 − 2 = x 4 So, the result is: 2 x 4
Each calculation uses basic properties of exponents to simplify the expression. Understanding these properties can help solve many problems related to exponentiation. Feel free to ask if you need further clarification!

Answered by danjohnbrain | 2025-07-07