Question

find the value of n if 8^n + 8^n+1= 72​

Answer 1

$\large\text{\underline{Exponent laws(Integers/Rationals/Reals)}}$

$\begin{gathered}\boxed{\begin{minipage}{8 cm}\underline{Laws of exponents(Integer exponents)} \\ \\For a\neq0 and b\neq0 and integers x and y , \\ \\\bigstar a^{0}=1 and a^{-n}=\dfrac{1}{a^{n}} \\ \\ \bullet a^{x} \times a^{y}=a^{x+y} \\ \\\bullet {a}^{x} \div {a}^{y}={a}^{x-y} \\ \\\bullet ({a}^{x})^{y} =a^{xy} \\ \\\bullet (ab)^{x}=a^{x}b^{x} \end{minipage}}\end{gathered}$

$\begin{gathered}\boxed{\begin{minipage}{8 cm}\underline{Laws of exponents(Rational exponents)} \\ \\For a\neq0 and b\neq0 and rationals r and s , \\ \\ \bigstar a^{\frac{1}{n}}=\sqrt[n]{a} and a^{\frac{m}{n}}=\sqrt[n]{a^{m}} \\ \\\bullet a^{r} \times a^{s}=a^{r+s} \\ \\\bullet {a}^{r} \div {a}^{s}={a}^{r-s} \\ \\\bullet ({a}^{r})^{s} =a^{rs} \\ \\\bullet (ab)^{r}=a^{r}b^{r} \end{minipage}}\end{gathered}$

$\begin{gathered}\boxed{\begin{minipage}{8 cm}\underline{Laws of exponents(Real exponents)} \\ \\For a\neq0 and b\neq0 and reals x and y , \\ \\ \bullet a^{x} \times a^{y}=a^{x+y} \\ \\\bullet {a}^{x} \div {a}^{y}={a}^{x-y} \\ \\\bullet ({a}^{x})^{y} =a^{xy} \\ \\\bullet (ab)^{x}=a^{x}b^{x} \end{minipage}}\end{gathered}$

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

We suppose $n$ as a real number.

By exponent laws,

$\cdots\longrightarrow 8^{n+1}=8\cdot8^{n}$

The given equation is

$\cdots\longrightarrow 8^{n}+8\cdot 8^{n}=72$

$\cdots\longrightarrow 9\cdot 8^{n}=72$

$\cdots\longrightarrow 8^{n}=8$

$\cdots\longrightarrow n=1$

$\large\text{\underline{Answer}}$

$n=1$ is the only real solution to this equation.

Answer 2

Answer:

n= 1

Step-by-step explanation:

Solution

We suppose nn as a real number.

By exponent laws,

\cdots\longrightarrow 8^{n+1}=8\cdot8^{n}⋯⟶8

n+1

=8⋅8

n

The given equation is

\cdots\longrightarrow 8^{n}+8\cdot 8^{n}=72⋯⟶8

n

+8⋅8

n

=72

\cdots\longrightarrow 9\cdot 8^{n}=72⋯⟶9⋅8

n

=72

\cdots\longrightarrow 8^{n}=8⋯⟶8

n

=8

\cdots\longrightarrow n=1⋯⟶n=1

\large\text{\underline{Answer}}

Answer

n=1n=1 is the only real solution to this equation.