Question

Fill in the blanks.

(i) If 9 × 3ⁿ = 3⁶, then n = ...... .
(ii) (8)⁰ = ?
(iii) ${\left(\frac{a}{b}\right)}^{-16}=......$
(iv) (−2)⁻⁵ = ......

Answer 1

(i) If 9 × 3ⁿ = 3⁶, then n = 4.
Explanation:
If 9 × 3ⁿ = 3⁶
⇒ 3² × 3ⁿ = 3⁶
⇒ 3^{(2 + n)} = 3⁶
Equating the powers:
⇒ ( 2 + n) = 6
⇒ n = (6 - 2) = 4

(ii) (8)⁰ = 1
Explanation:
By definition, we have a⁰ = 1 for every integer a.
∴ (8)⁰ = 1

(iii) ${\left(\frac{a}{b}\right)}^{-16}$ = ${\left(\frac{b}{a}\right)}^{16}$
Explanation:
We know:
${\left(\frac{a}{b}\right)}^{-1}={\left(\frac{b}{a}\right)}^1$

(iv) (−2)⁻⁵ = $\frac{-1}{32}$
Explanation:
(−2)⁻⁵ = ${\left(\frac{-2}{1}\right)}^{-5}={\left(\frac{1}{-2}\right)}^5$ $\left[\mathrm{Since}{\left(\frac{a}{b}\right)}^{-1}={\left(\frac{b}{a}\right)}^1\right]$
= $\frac{{\left(1\right)}^5}{{\left(-2\right)}^5}=\frac{1\times 1\times 1\times 1\times 1}{\left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)\times \left(-2\right)}=\frac{1}{-32}$