Question

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Simplify and express each as a rational number :

(i) $\sf\large \bigg(\frac{4}{9}\bigg)^6 \: \times \: \sf\large \bigg(\frac{4}{9}\bigg)^{-4}$

(ii) $\sf\large \bigg(\frac{-7}{8}\bigg)^{-3} \: \times \: \sf\large \bigg(\frac{-7}{8}\bigg)^{2}$

(iii) $\sf\large \bigg(\frac{4}{3}\bigg)^{-3} \: \times \: \sf\large \bigg(\frac{4}{3}\bigg)^{-2}$
​

Answer 1

Answer:

(i) 16/81

(ii) 8/7

(iii) 243/1024

Step-by-step explanation:

I am not using LaTeX to save time. Sorry for the inconveniences !

(i) (4/9)⁶ × (4/9)⁻⁴

= 4⁶/9⁶ × (9/4)⁴

= 4⁶/9⁶ × 9⁴/4⁴

= 4⁶⁻⁴ × 9⁴⁻⁶

= 4² × 9⁻²

= 4²/9²

= 16/81

(ii) (-7/8)⁻³ × (-7/8)²

= (8/-7)³ × (-7/8)²

= 8³/-7³ × -7²/8²

= 8³⁻² × -7²⁻³

= 8 × -7⁻¹

= 8/7

(iii) (4/3)⁻³ × (4/3)⁻²

= (3/4)³ × (3/4)²

= 3³/4³ × 3²/4²

= 3³⁺²/4³⁺²

= 3⁵/4⁵

= (3/4)⁵

= 243/1024

LAWS OF EXPONENT :

1. aᵐ × aⁿ = aᵐ ⁺ ⁿ
2. aᵐ ÷ aⁿ = aᵐ ⁻ ⁿ
3. a⁻ᵐ = 1/aᵐ
4. a⁻¹ = 1/a
5. (ab)ᵐ = aᵐ × bᵐ
6. (a/b)ᵐ = aᵐ / bᵐ
7. a⁰ = 1

Thanks !

Answer 2

QUESTION :- Simplify and express each as a rational number :-

1•} $(\frac{4}{9})^6 \: \times (\frac{4}{9})^{-4}$

$= \frac{ {4}^{6} }{ {9}^{6} } \times (\frac{9}{4})^{4}$

$= \frac{ {4}^{6} }{ {9}^{6} } \times \frac{ {9}^{4} }{ {4}^{4} }$

$={4}^{6 - 4} \times {9}^{4 - 6}$

$= {4}^{2} \times {9}^{ - 2}$

$= \frac{ {4}^{2} }{ {9}^{2} }$

$= \frac{16}{81}$

2•} $(\frac{-7}{8})^{-3} \times (\frac{-7}{8})^{2}$

$= {(\frac{8}{ - 7} })^{3} \times { (\frac{ - 7}{8}) }^{2}$

$= \frac{ {8}^{3} }{ { - 7}^{3} } \times \frac{ { - 7}^{2} }{ {8}^{2} }$

$= {8}^{3 - 2} \times { - 7}^{2 - 3}$

$= 8 \times { - 7}^{ - 1}$

$= \frac{8}{ - 7}$

3•} $(\frac{4}{3})^{-3} \times(\frac{4}{3})^{-2}$

$= {( \frac{3}{4} )}^{3} \times {( \frac{3}{4}) }^{2}$

$= \frac{ {3}^{3} }{ {4}^{3} } \times \frac{ {3}^{2} }{ {4}^{2} }$

$= \frac{ {3}^{3 + 2} }{ {4}^{3 + 2} }$

$= \frac{ {3}^{5} }{ {4}^{5} }$

$= {( \frac{3}{4}) }^{5}$

$= \frac{243}{1024}$

$\huge\fbox\pink{\sf @IIMrVelvetII}$

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