[tex]\Large{\underbrace{\sf{\orange{Question:}}}}[/tex]
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[tex]\large{\underline{\bf{Source}}}[/tex] - Class 9th, R.D. Sharma Mathematics Book (CBSE).
[tex]\large{\underline{\bf{Chapter}}}[/tex] - 1, Number System.
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[tex]\Large{\underbrace{\underline{\bf{QUESTION\:}}}}: \\ [/tex]
Find the value of 'p'.
[tex]\dagger\:\bf\red{5^{(p\:-\:5)}\times{2^{(p\:-\:5)}}\:=\:5\:} \\ \\ [/tex]
[tex]\Large{\underbrace{\underline{\bf{ANSWER\:}}}}: \\ [/tex]
[tex]{\bf{Given\::}} \\ [/tex]
[tex]\red\checkmark\:\tt{5^{(p\:-\:5)}\times{2^{(p\:-\:5)}}\:=\:5\:} \\ [/tex]
[tex] \\ {\bf{To\: Find\::}} \\ [/tex]
[tex] \\ {\bf{Properties\::}} \\ [/tex]
[tex]\dagger\:\bf\blue{a^{1}\:=\:a\:} \\ [/tex]
[tex]\dagger\:\bf\purple{a^{0}\:=\:1\:} \\ [/tex]
[tex] \\ {\bf{Solution\::}} \\ [/tex]
[tex]:\implies\:\tt{5^{(p\:-\:5)}\times{2^{(p\:-\:5)}}\:=\:5\:} \\ [/tex]
☛ We also write as,
[tex]:\implies\:\tt{5^{(p\:-\:5)}\times{2^{(p\:-\:5)}}\:=\:5\times{1}\:} \\ [/tex]
[tex]:\implies\:\tt{5^{(p\:-\:5)}\times{2^{(p\:-\:5)}}\:=\:5^{1}\times{2^{0}}\:} \\ [/tex]
☛ Compare both sides, we get
[tex]:\implies\:\tt{5^{(p\:-\:5)}\:=\:5^{1}\:~~~\&\:~~~\:{2^{(p\:-\:5)}}\:=\:2^{0}\:} \\ [/tex]
[tex]:\implies\:\tt{p\:-\:5\:=\:1\:~~~Or\:~~~\:p\:-\:5\:=\:0\:} \\ [/tex]
[tex]:\implies\:\tt{p\:=\:1\:+\:5\:~~~Or\:~~~\:p\:=\:0\:+\:5\:} \\ [/tex]
[tex]:\implies\:{\tt{\green {p\:=\:6}}}\:~~~Or\:~~~\:{\tt{\green {p\:=\:5}}}\: \\ [/tex]
[tex]\large{\underline{\underline{\red{\bf{Given}}}}}[/tex]
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[tex]\large{\underline{\underline{\red{\bf{To\: find}}}}}[/tex]
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[tex]\large{\underline{\underline{\red{\bf{Solution}}}}}[/tex]
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As we can see that, right and side can be written as 5 × 1
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[tex]\tt\dashrightarrow{5^{(p - 5)} \times 2^{(p - 5)} = 5 \times 1}[/tex]
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Now, let's see some identities used here
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Reminding the both identities, we can write
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[tex]\tt\dashrightarrow{5^{(p - 5)} \times 2^{(p - 5)} = 5^1 \times 2^0}[/tex]
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Comparing both the sides
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[tex]\tt:\implies{p - 5 = 1}[/tex]
[tex]\tt:\implies{p = 1 + 5}[/tex]
[tex]\tt:\implies{p = 6}[/tex]
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Similarly,
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[tex]\tt:\implies{p - 5 = 0}[/tex]
[tex]\tt:\implies{p = 5}[/tex]
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Hence,
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[tex]\: \: \: \: \: \: \: \: \: \: \: \: \boxed{\bf{\bigstar{More\: to\: know{\bigstar}}}}[/tex]
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