Find the value of 'a' satisfying the equation. a) (42)a=(7a)2. b) (2a)

Question

Find the value of 'a' satisfying the equation.
a)

{(4^{2})}^{a} = {(7^{a})}^{2}

.
b)

(2^{a})^{5} = 2^{4} \times 4^{3}

[4 MARKS]

Answer:
:
Each option: 2 Marks
a) Given that,

{(4^{2})}^{a} = {(7^{a})}^{2}

4^{2 a} = 7^{2 a}

[{a^{m}}^{n} = a^{m n}]

Given,

4^{2 a} = 7^{2 a}

Since

4 \neq 7

the only case where the above equation is true is when both the exponents are zero.

\Rightarrow a = 0

\Rightarrow 4^{2 a} = 7^{2 a} = 1

[∵ 4^{0} = 1, 7^{0} = 1]

b) The given equation is

(2^{a})^{5} = 2^{4} \times 4^{3}

\Rightarrow 2^{a \times 5} = 2^{4} \times (2^{2})^{3}

∵ (a^{m})^{n} = a^{m \times n}

\Rightarrow 2^{a \times 5} = 2^{4} \times (2^{2 \times 3})

\Rightarrow 2^{a \times 5} = 2^{4} \times (2^{6})

\Rightarrow 2^{a \times 5} = 2^{4 + 6} = 2^{10}

∵ a^{m} \times a^{n} = a^{m + n}

Since their bases are same andthey are equal, threforetheir powers must be same,
So,

5 \times a = 10

\Rightarrow a = \frac{10}{5}

\Rightarrow a = 2

So, the value of a = 2.

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