Question

$\red{\tt \: If{\Bigg[ \sqrt[9]{ \bigg( \frac{2}{3}\bigg)^{5} }\Bigg]^{ \sqrt{x - 5}} = a^{0},\:then \: find \: the \: value \: of \: x.}}$
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Answer 1

⚘Question:-

${\tt \: If{\Bigg[ \sqrt[9]{ \bigg( \frac{2}{3}\bigg)^{5} }\Bigg]^{ \sqrt{x - 5}} = {a}^{0} }}$

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⚘To find it:-

• Find the value of x.

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⚘Answer

$\bold{(a \neq \: 0)}$

${\tt \: {\Bigg[ \sqrt[9]{ \bigg( \frac{2}{3}\bigg)^{5} }\Bigg]^{ \sqrt{x - 5}} = a^{0} = 1}}$

$\bold{\implies \: {\tt \:{\ \bigg[ { \bigg( \frac{2}{3}\bigg)^{ \frac{5}{9} } }\bigg]^{ \sqrt{x - 5}} = 1{}}}}$

$\bold{(aᵐ)ⁿ = aᵐⁿ}$

$\mathbb{\implies( \frac{2}{3}) \frac{5}{5} \sqrt{a - 5} = 1 }$

$\bold{ \implies \: ( \frac{2}{3} )⁰}$

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$\mathsf\purple{❤|•ItzShizuka50•|❤}$

Answer 2

$\huge \red{ \boxed{ \fcolorbox{blue}{magenta}{ \bf \orange{x} = \green{5}}}}$

Step-by-step explanation:

QUESTION :-

$\tt \: If{\Bigg[ \sqrt[9]{ \bigg( \frac{2}{3}\bigg)^{5} }\Bigg]^{ \sqrt{x - 5}} = a^{0}}, \\ \tt\:then \: find \: the \: value \: of \: x.$

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SOLUTION :-

We know that,

any number's power raised to 0 = 1

So,

a⁰ = 1

$\bf = > {\Bigg[ \sqrt[9]{ \bigg( \frac{2}{3}\bigg)^{5} }\Bigg]^{ \sqrt{x - 5}} = a^{0}} \\ \\ \bf = > \Bigg[ \sqrt[9]{ \bigg( \frac{2}{3}\bigg)^{5} }\Bigg]^{ \sqrt{x - 5}} =1$

$\bf = >{\Bigg[ \bigg( \frac{2}{3}\bigg)^{ \frac{5}{9}} }\Bigg]^{ \sqrt{x - 5}} = 1... \tiny \{{using \: \sqrt[m]{ {x}^{n} } = {x}^{ \frac{n}{m} } } \}$

$\bf = > \bigg( \frac{2}{3} \bigg) {}^{ \frac{5}{9} \times \sqrt{x - 5} } = 1... \tiny{ \{using {(x {}^{m}) }^{n} = {x}^{m \times n} \}}$

As mentioned above,

any number's power 0 = 1

Opposite of this statement can be,

1 = any number's power 0

$= > 1 = ( \frac{2}{3} {)}^{0}$

So,

$\bf = > \bigg( \frac{2}{3} \bigg) {}^{ \frac{5 \sqrt{x - 5} }{9} } = \bigg( \frac{2}{3} { \bigg)}^{0}$

$\bf = > \frac{5 \sqrt{x - 5} }{9} = 0... \tiny{ \{using \: {x}^{m} = {x}^{n} = > m = n \}} \\ \\ \bf = > 5 \sqrt{x - 5} = 0 \times 9 \\ \\ \bf = > \sqrt{x - 5} = \frac{0}{5} \\ \\ \bf = > \sqrt{x - 5} = 0 \\ \\ \{by \: squaring \: both \: sides \} \\ \\ \bf = > x - 5 = 0 \\ \\ = > \large \fcolorbox{red}{pink}{ \bf \blue{x = 5}}$

Hence,

The value of x = 5.

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VERIFICATION :-

x = 5

[any number's power raised to 0 = 1]

$\bf = >\Bigg[ \sqrt[9]{ \bigg( \frac{2}{3}\bigg)^{5} }\Bigg]^{ \sqrt{x - 5}} = a^{0} \\ \\ \bf = > \Bigg[ \sqrt[9]{ \bigg( \frac{2}{3}\bigg)^{5} }\Bigg]^{ \sqrt{5 - 5}} = 1 \\ \\ \bf = > \Bigg[ \sqrt[9]{ \bigg( \frac{2}{3}\bigg)^{5} }\Bigg]^{0} = 1 \\ \\ \bf = > 1 = 1 \\ \\ \bf = > LHS = RHS$

Hence Verified.

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Hope it helps.

#BeBrainly :-)