By what number should (-3/2)⁻³ be divided so that the quotient may be (4/27)⁻²?

Solution:

Given, (-3/2)⁻³ should be divided by a number so that the quotient may be (4/27)⁻².

We have to find the number.

Let the number be x.

According to the question,

(-3/2)⁻³ ÷ x = (4/27)⁻²

(-3/2)⁻³ × 1/x = (4/27)⁻²

Solving for x,

x = (-3/2)⁻³ / (4/27)⁻²

Considering (4/27)⁻²,

(4/27)⁻² can be written as (2²/3³)⁻²

(a^m)ⁿ= a^mn

So, [(2²/3³)]⁻² = (2⁻⁴/3⁻⁶)

Now, x = (-3/2)⁻³ / (2⁻⁴/3⁻⁶)

Using law of exponents,

a⁻ⁿ = 1/aⁿ

So, (-3/2)⁻³ = 1/(-3/2)⁻³

= (-2/3)³

Similarly, (2⁻⁴/3⁻⁶) = 1/(2⁻⁴/3⁻⁶)

= (3⁶/2⁴)

Now, x = (-2/3)³ / (3⁶/2⁴)

= (-2/3)³ × (2⁴/3⁶)

= (-2)³ × (2)⁴ / (3)³ × (3)⁶

Using law of exponents,

a^m × aⁿ = a^{m + n}

= -(2)^{4 + 3}/(3)^{3 + 6}

= (-2)⁷/(3)⁹

Therefore, the required value is (-2)⁷/(3)⁹.

✦ Try This: By what number should (2/4)⁻³ be divided so that the quotient may be (4/16)⁻²?

☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 12

NCERT Exemplar Class 8 Maths Chapter 8 Problem 117

Summary:

(-3/2)⁻³ should be divided by (-2)⁷/(3)⁹ so that the quotient may be (4/27)⁻².

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