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Evaluate : (15⁴ ×18³)/(3³ × 5² ×12²)

Solution:

Given, the expression is (15⁴ ×18³)/(3³ × 5² ×12²)

We have to evaluate the expression

15⁴ can be written as (3 × 5)⁴

18³ can be written as (2 × 9)³

= (2 × 3²)³

12² can be written as (4 × 3)²

= (2² × 3)²

For any non-zero integers ‘a’ and ‘b’ and whole numbers m and n,

a^m × b^m = (a×b)^m

So, (3 × 5)⁴ = 3⁴ × 5⁴

(2 × 3²)³ = 2³ × (3²)³

(2² × 3)² = (2²)² × 3²

For any non-zero integer ‘a’ and whole numbers m and n,

(a^m)ⁿ = (a)^mn

So, (2²)² = 2^{2 × 2} = 2⁴

(3²)³ = 3⁶

Now, (15⁴ ×18³) = 3⁴ × 5⁴ × 2³ × 3⁶

For any non-zero integers ‘a’ and ‘b’ and whole numbers m and n,

a^m × aⁿ = a^m+n

3⁴ × 3⁶ = 3¹⁰

So, 3⁴ × 5⁴ × 2³ × 3⁶ = 3¹⁰ × 5⁴ × 2³

Similarly, 3³ × 5² ×12² = 3² × 5² × 2⁴ × 3²

= 3⁵ × 5² × 2⁴

The expression (15⁴ ×18³)/(3² × 5² ×12²) becomes (3¹⁰ × 5⁴ × 2³)/(3⁵ × 5² × 2⁴)

For any non-zero integers ‘a’ and ‘b’ and whole numbers m and n,

a^m ÷ aⁿ = a^m-n

Considering 3¹⁰/3⁵,

Here, a = 3, m = 10 and n = 5

m - n = 10 - 5 = 5

So, 3¹⁰/3⁴ = 3⁵

= 243

Similarly, 5⁴/5² = 5² = 25

2³/2⁴ = 2⁻¹ = 1/2

Now, (15⁴ ×18³)/(3² × 5² ×12²) = 243×25×1/2

= 6075/2

Therefore, the required value is 6075/2

✦ Try This: Evaluate : (25⁴ ×36²)/(5³ × 6²)

☛ Also Check: NCERT Solutions for Class 7 Maths Chapter 13

NCERT Exemplar Class 7 Maths Chapter 11 Problem 85 (f)

Summary:

On evaluating (15⁴ ×18³)/(3² × 5² ×12²) we get 6075/2

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