Question

If (25ᵃ) (5)ᵃ⁺⁴ (125)²ᵃ = (625)¹⁰
Find a?

Answer 1

Given

• (25ᵃ) (5)ᵃ⁺⁴ (125)²ᵃ = (625)¹⁰

To find

• Value of a

Solution

• (25ᵃ) (5)ᵃ⁺⁴ (125)²ᵃ = (625)¹⁰
• (5²)ᵃ (5)ᵃ⁺⁴ (5³)²ᵃ = (5⁴)¹⁰
• 5²ᵃ × 5ᵃ⁺⁴ × 5⁶ᵃ = 5⁴⁰

☯ Bases are same, add the power

• 5²ᵃ⁺ᵃ⁺⁴⁺⁶ᵃ = 5⁴⁰
• 5⁹ᵃ⁺⁴ = 5⁴⁰

☯ Bases are equal, equate the power

• 9a+4 = 40
• 9a = 40-4
• 9a = 36
• a = ³⁶⁄₉
• a = 4

☯ Value of a = 4

Answer 2

Given : -

$\tt \:  {(25)}^{a} {(5)}^{a + 4} {(125)}^{2a} = {(625)}^{10}$

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$\begin{gathered}\begin{gathered}\bf  To \:  Find :-  \begin{cases} &\sf{value \: of \: a}  \end{cases}\end{gathered}\end{gathered}$

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$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

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$\begin{gathered}(1)\:{\underline{\boxed{\bf{\blue{a^m\times{a^n}\:=\:a^{m\:+\:n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(2)\:{\underline{\boxed{\bf{\blue{a^m \: = \: {a^n}\: \implies\:{m\: = \:n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(3)\:{\underline{\boxed{\bf{\color{peru}{(a^m)^n\:=\:a^{m\times{n}}\:}}}}} \\ \end{gathered}$

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$\large\underline\purple{\bold{Solution :- }}$

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$\tt \:  \longrightarrow \: {(25)}^{a} {(5)}^{a + 4} {(125)}^{2a} = {(625)}^{10}$

$\tt \:  \longrightarrow \: {( 5\times 5)}^{a} {(5)}^{a + 4} {(5 \times 5 \times 5)}^{2a} = {(5 \times 5 \times 5 \times 5)}^{10}$

$\tt \:  \longrightarrow \: {(5)}^{2a} {(5)}^{a + 4} {(5)}^{6a} = {(5)}^{40}$

$\tt \:  \longrightarrow \: {(5)}^{2a + a + 4 + 6a} = {(5)}^{40}$

$\tt \:  \longrightarrow \: {(5)}^{9a + 4} = {5}^{40}$

$\tt\implies \:9a + 4 = 40$

$\tt \:  \longrightarrow \: 9a = 40 - 4$

$\tt \:  \longrightarrow \: 9a = 36$

$\tt\implies \:a \: = \: 4$

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$\large{\boxed{\boxed{\bf{Hence, value \: of \: a \: = \: 4}}}}$

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