Question

$\Large{\underbrace{\sf{\orange{Question:}}}}$
[ Refer to the attached picture! :) ]

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$\large{\underline{\bf{Source}}}$ - Class 9th, R.D. Sharma Mathematics Book (CBSE).

$\large{\underline{\bf{Chapter}}}$ - 1, Number System.




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Answer 1

$\Large{\underbrace{\underline{\bf{QUESTION\:}}}}:$

Find the value of 'p'.

$\dagger\:\bf\red{5^{(p\:-\:5)}\times{2^{(p\:-\:5)}}\:=\:5\:}$

$\Large{\underbrace{\underline{\bf{ANSWER\:}}}}:$

${\bf{Given\::}}$

$\red\checkmark\:\tt{5^{(p\:-\:5)}\times{2^{(p\:-\:5)}}\:=\:5\:}$

$\\ {\bf{To\: Find\::}}$

• The value of 'p'.

$\\ {\bf{Properties\::}}$

$\dagger\:\bf\blue{a^{1}\:=\:a\:}$

$\dagger\:\bf\purple{a^{0}\:=\:1\:}$

$\\ {\bf{Solution\::}}$

$:\implies\:\tt{5^{(p\:-\:5)}\times{2^{(p\:-\:5)}}\:=\:5\:}$

☛ We also write as,

$:\implies\:\tt{5^{(p\:-\:5)}\times{2^{(p\:-\:5)}}\:=\:5\times{1}\:}$

$:\implies\:\tt{5^{(p\:-\:5)}\times{2^{(p\:-\:5)}}\:=\:5^{1}\times{2^{0}}\:}$

☛ Compare both sides, we get

$:\implies\:\tt{5^{(p\:-\:5)}\:=\:5^{1}\:~~~\&\:~~~\:{2^{(p\:-\:5)}}\:=\:2^{0}\:}$

$:\implies\:\tt{p\:-\:5\:=\:1\:~~~Or\:~~~\:p\:-\:5\:=\:0\:}$

$:\implies\:\tt{p\:=\:1\:+\:5\:~~~Or\:~~~\:p\:=\:0\:+\:5\:}$

$:\implies\:{\tt{\green {p\:=\:6}}}\:~~~Or\:~~~\:{\tt{\green {p\:=\:5}}}\:$

Answer 2

$\large{\underline{\underline{\red{\bf{Given}}}}}$

• $\sf{5^{(p - 5)} \times 2^{(p - 5)} = 5}$

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$\large{\underline{\underline{\red{\bf{To\: find}}}}}$

• Value of p.

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$\large{\underline{\underline{\red{\bf{Solution}}}}}$

• We have an exponential equation and need to find out the value of p.

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As we can see that, right and side can be written as 5 × 1

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$\tt\dashrightarrow{5^{(p - 5)} \times 2^{(p - 5)} = 5 \times 1}$

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Now, let's see some identities used here

• $\sf{a^1 = a}$
• $\sf{a^0 = 1}$

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Reminding the both identities, we can write

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$\tt\dashrightarrow{5^{(p - 5)} \times 2^{(p - 5)} = 5^1 \times 2^0}$

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Comparing both the sides

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$\tt:\implies{p - 5 = 1}$

$\tt:\implies{p = 1 + 5}$

$\tt:\implies{p = 6}$

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Similarly,

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$\tt:\implies{p - 5 = 0}$

$\tt:\implies{p = 5}$

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Hence,

• Value of p is 5 or 6.

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$\: \: \: \: \: \: \: \: \: \: \: \: \boxed{\bf{\bigstar{More\: to\: know{\bigstar}}}}$

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• $\sf{a^n \times a^m = a^{n + m}}$

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• $\sf{a^n \div a^m = a^{n - m}}$

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• $\sf{a^n \times b^n = ab^n}$

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• $\sf{(a^n)^m = a^{nm}}$