Question

Simplify and express the result in power notation with positive exponent.

1. (−4) ⁵ ÷ (−4) ⁸ 
2. \(\left(\frac{1 }{ 2^3}\right)\) ² 
3. (−3) ⁴ × (\(\frac{5}{3}\)) ⁴
4. (3 ^-7 ÷ 3 ^-10) × 3 ^-5 
5. 2 ^-3 × (−7) ^-3

Answer 1

(i) (−4)⁵ ÷ (−4)⁸
Since \(\frac{a^m}{a^n}\)= a^{m − n} 
(−4)⁵ ÷ (−4)⁸ = \(\frac{(−4)^5}{(−4)^8}\)= (−4)⁵⁻⁸
(−4)⁻³
\(= \frac{1}{(−4)^3}\)

---

(ii) \(\left(\frac{1}{2^3}\right)^2\)
Since, (a^m)ⁿ = a^mn
\(⇒ \left(\frac{1}{2^3}\right)^2\)
\(= \frac{1}{2^6}\)

---

(iii) (−3)⁴ \(× \left(\frac{5}{3}\right)^4\)
We know that a^m × b^m =(ab)^m and (\(\frac{a}{b}\))^m = \(\frac{a^m}{b^m}\) where a & b are non-zero integers and m is an integer
\(⇒\) (−3)⁴ \(× \left(\frac{5}{3}\right)^4\)

\(⇒ (−1 × 3)^4 × \frac{5^4}{3^4}\)

\(⇒\) (−1)⁴ × 5⁴
= 5⁴      [∵(−1)⁴ = 1]

---

(iv) (3⁻⁷ ÷ 3⁻¹⁰) × 3⁻⁵
We know.  \(\frac{a^m}{a^n} =\) a^{m − n} and a^m × aⁿ = a^{m + n}
(3⁻⁷ ÷ 3⁻¹⁰) × 3⁻⁵ = (3^{−7−(−10)}) × 3⁻⁵
= (3^{−7 + 10}) × 3⁻⁵ = 3³ × (3⁻⁵)
= 3^{3 + (−5)}
= 3⁻² =\( \frac{1}{3^2}\)

---

(v) 2⁻³ × (−7)⁻³
We know that, a^m × b^m = (ab)^m
2⁻³ × (−7)⁻³
= [2 × (−7)]⁻³
= (−14)⁻³
\(= \frac{1}{(-14)^3}\)   [Since a^−m = 1/a^m]