Exponents and Powers — Mental Maths

Fill in the blanks:

(i) Multiplicative inverse of $\Big(\dfrac{-4}{5}\Big)^2$ = ......

(ii) If $(-7)^x \times \Big(\dfrac{1}{-7}\Big)^{-3} = 7^8$, then value of x is ......

(iii) If 3ⁿ = 729, then n = ......

(iv) The value of (7⁰ + 5⁰) is ......

Answer

(i) Solving,

$\Rightarrow \Big(\dfrac{-4}{5}\Big)^2\\[1em] \Rightarrow \dfrac{(-4) \times (-4)}{5 \times 5}\\[1em] \Rightarrow \dfrac{16}{25}$

The multiplicative inverse of a non-zero number is its reciprocal.

Thus, multiplicative inverse of $\dfrac{16}{25} = \dfrac{25}{16}$.

Hence, multiplicative inverse of $\mathbf{\Big(\dfrac{-4}{5}\Big)^2} = \mathbf{\dfrac{25}{16}}$.

(ii) As 8 is an even number and a negative number raised to an even power gives a positive value,

$\Rightarrow (-7)^8\\[1em] \Rightarrow (-1 \times 7)^8\\[1em] \Rightarrow (-1)^8 \times 7^8\\[1em] \Rightarrow 1 \times 7^8\\[1em] \Rightarrow 7^8$

Thus, 7⁸ = (-7)⁸.

Solving,

$\Rightarrow (-7)^x \times \Big(\dfrac{1}{-7}\Big)^{-3} = 7^8\\[1em] \Rightarrow (-7)^x \times (-7)^{3} = (-7)^8\\[1em] \Rightarrow (-7)^{x + 3} = (-7)^8$

Using $a^m = a^n \Rightarrow m = n$

$\Rightarrow x + 3 = 8\\[1em] \Rightarrow x = 8 - 3\\[1em] \Rightarrow x = 5$

Hence, the value of x is 5.

(iii) We know that 729 = 3 x 3 x 3 x 3 x 3 x 3 = 3⁶

$\Rightarrow 3^n = 729\\[1em] \Rightarrow 3^n = 3^6$

Using $a^m = a^n \Rightarrow m = n$

$\Rightarrow n = 6$

Hence, the value of n is 6.

(iv) We know that,

a⁰ = 1

⇒ 7⁰ + 5⁰

⇒ 1 + 1

⇒ 2

Hence, the value of (7⁰ + 5⁰) is 2.

State whether the following statements are true (T) or false (F):

(i) The value of (-3)^-2 is $\dfrac{1}{-9}$

(ii) For negative integers m and n, a^m × aⁿ = a^m+n (a ≠ 0)

(iii) 3⁰ × 4⁰ = 12⁰

(iv) The value of 3⁷ - 2⁷ is 1⁷

(v) Multiplicative inverse of (-2)⁵ is $\dfrac{1}{-32}$

Answer

(i) Solving,

$\Rightarrow (-3)^{-2}\\[1em] \Rightarrow \dfrac{1}{(-3)^2}\\[1em] \Rightarrow \dfrac{1}{(-3) \times (-3)}\\[1em] \Rightarrow \dfrac{1}{9}$

Since the value of (-3)^-2 is $\dfrac{1}{9}$ and not $\dfrac{1}{-9}$, the statement is false.

Hence, the given statement is False.

(ii) The law of exponents $a^m \times a^n = a^{m+n}$ holds for all integers m and n (positive, negative or zero), where a ≠ 0.

Since m and n are negative integers, this law is still valid.

Hence, the given statement is True.

(iii) We know that $a^0 = 1$

Solving L.H.S.,

⇒ 3⁰ × 4⁰

⇒ 1 × 1

⇒ 1

Solving R.H.S.,

⇒ 12⁰ = 1

Since, L.H.S. = R.H.S.

Hence, the given statement is True.

(iv) Solving L.H.S.,

⇒ 3⁷ - 2⁷

⇒ 2187 - 128

⇒ 2059

Solving R.H.S.,

⇒ 1⁷ = 1

Since, L.H.S. ≠ R.H.S.

Hence, the given statement is False.

(v) Solving,

⇒ (-2)⁵

⇒ (-2) x (-2) x (-2) x (-2) x (-2)

⇒ -32

The multiplicative inverse of -32 is $\dfrac{1}{-32}$.

Hence, the given statement is True.