Question

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Answer 1

$\large\text{\underline{Let's begin.}}$

We are given an exponential equation. To solve this equation, we need to use the property of exponent.

$\large\text{\underline{First to know?}}$

• $a^{m}\times a^{n}=a^{m+n}$

This is a property of exponents.

• Comparing the exponents.

This is a property of exponents for an equal base. If bases are equal and two powers are equal, the exponent is also equal.

$\large\text{\underline{Solution}}$

$\implies \left(\dfrac{5}{3}\right)^{-5}\times\left(\dfrac{5}{3}\right)^{-11}=\left(\dfrac{5}{3}\right)^{8x}$

$\implies \left(\dfrac{5}{3}\right)^{-5-11}=\left(\dfrac{5}{3}\right)^{8x}$

$\implies \left(\dfrac{5}{3}\right)^{-16}=\left(\dfrac{5}{3}\right)^{8x}$

$\implies 8x=-16\implies \therefore x=-2$

$\large\text{\underline{Conclusion}}$

Hence, the value of $x$ is -2. This is the end of the required answer.

$\large\text{\underline{Advanced Question}}$

Here is an equation of example, find the value of $x$.

$\hookrightarrow\left(2+\sqrt{3}\right)^{x}+\left(2-\sqrt{3}\right)^{x}=2$

$\large\text{\underline{First to know?}}$

• $a^{0}=1\ (a\neq0)$

If the exponent 0 is applied on non-zero numbers, the power becomes 1.

$\large\text{\underline{Solution}}$

We can note that the two bases are reciprocal.

So, we can substitute $t=\left(2+\sqrt{3}\right)^{x}$.

Then $\dfrac{1}{t}=\left(2-\sqrt{3}\right)^{x}$.

$\implies t+\dfrac{1}{t}=2$

$\implies t^{2}-2t+1=0\implies\therefore t=1$

After this, substitute $t$ with the given value,

$\implies \left(2+\sqrt{3}\right)^{x}=1$

We learned that $a^{0}=1\ (a\neq0)$.

$\implies \left(2+\sqrt{3}\right)^{x}=\left(2+\sqrt{3}\right)^{0}$

Comparing the exponent,

$\implies x=0$

$\large\text{\underline{Conclusion}}$

Hence, the value of $x$ is 0.

Answer 2

Answer:

$\sf\tt\large{\green {\underline {\underline{⚘\;Question:}}}}$

• Find x so that,

• $( \frac{5}{3}) {}^{ - 5} \times ( \frac{5}{3} ) {}^{ - 11} = (\frac{5}{3} ) {}^{8x}$

$\sf\tt\large{\green {\underline {\underline{⚘\;Answer:}}}}$

$\sf\tt\large{\purple {\underline {\underline{⚘\;The \;value \;of \;x \;is \;-2:}}}}$

$\sf\tt\large{\red {\underline {\underline{⚘\;Given:}}}}$

• In the question given some exponential equation.

$\sf\tt\large{\red {\underline {\underline{⚘\;To find:}}}}$

• The value of x for given exponential equation.

$\sf\tt\large{\red {\underline {\underline{⚘\;Solution:}}}}$

• Here we know the one thing that is,

• ${a}^{m} \times {a}^{n} = {a}^{m + n}$
• (The property of exponents).

• According to exponents property lets apply the values.

• (5/3)^-5 ×(5/3)^-11= (5/3)^8x

• By solving this we get that,

• (5/3)^-11-5 = (5/3)^8x

• (5/3)^-16 = (5/3)^8x

• -16 = 8x

• -16/8 = x

• -2 = x.

$\sf\tt\large{\red {\underline {\underline{⚘\;Hope \;it \;helps \;you \;mate:}}}}$

$\sf\tt\large{\red {\underline {\underline{⚘\;Thank \;you:}}}}$