Solve $9^{\displaystyle x+4}$ $\,=\,$ $3^{\displaystyle 2} \times 27^{\displaystyle x+1}$

In this problem, it is given that the number $9$ raised to the power of $x$ plus $4$ is equal to $3$ squared times $27$ raised to the power of $x$ plus $1$.

$9^{\displaystyle x+4}$ $\,=\,$ $3^{\displaystyle 2} \times 27^{\displaystyle x+1}$

The value of $x$ has to evaluate by solving the given exponential equation in this exponential equation problem.

Basic method

Split each exponential function

There is an exponential function in terms of $x$ on the left-hand side of the equation. Similarly, there is another exponential function in terms of $x$ on the right-hand side of the equation. The two exponential functions are formed by the sum of a variable $x$ and a number. For solving the value of $x$, both exponential functions should be split. It can be done by the product rule of exponents.

$(1).\,\,\,$ $9^{\displaystyle x+4}$ $\,=\,$ $9^{\displaystyle x} \times 9^{\displaystyle 4}$

$(2).\,\,\,$ $27^{\displaystyle x+1}$ $\,=\,$ $27^{\displaystyle x} \times 27^{\displaystyle 1}$

Now, substitute them in the given exponential equation.

$\implies$ $9^{\displaystyle x} \times 9^{\displaystyle 4}$ $\,=\,$ $3^{\displaystyle 2} \times 27^{\displaystyle x} \times 27^{\displaystyle 1}$

$\implies$ $9^{\displaystyle x} \times 9^{\displaystyle 4}$ $\,=\,$ $3^{\displaystyle 2} \times 27^{\displaystyle x} \times 27$

d

On the left-hand side of the equation, the bases of both factors are $9$. On the right-hand side of the equation, the base of one factor is $3$ and the bases of the remaining factors are $27$. By factoring, the quantities can be expressed in exponential form with the base of $3$.

$(1).\,\,\,$ $9$ $\,=\,$ $3 \times 3$ $\,=\,$ $3^{\displaystyle 2}$

$(2).\,\,\,$ $27$ $\,=\,$ $3 \times 3 \times 3$ $\,=\,$ $3^{\displaystyle 3}$

Now, substitute them in the exponential equation.

$\implies$ $\Big(3^{\displaystyle 2}\Big)^{\displaystyle x} \times \Big(3^{\displaystyle 2}\Big)^{\displaystyle 4}$ $\,=\,$ $3^{\displaystyle 2} \times \Big(3^{\displaystyle 3}\Big)^{\displaystyle x} \times 3^{\displaystyle 3}$

The power of an exponential form quantity can be simplified by the power rule of exponents.

$\implies$ $\Big(3^{\displaystyle 2 \times x}\Big) \times \Big(3^{\displaystyle 2 \times 4}\Big)$ $\,=\,$ $3^{\displaystyle 2} \times \Big(3^{\displaystyle 3 \times x}\Big) \times 3^{\displaystyle 3}$

$\implies$ $3^{\displaystyle 2x} \times 3^{\displaystyle 8}$ $\,=\,$ $3^{\displaystyle 2} \times 3^{\displaystyle 3x} \times 3^{\displaystyle 3}$

Simplify the exponential equation

$\implies$ $3^{\displaystyle 2x} \times 3^{\displaystyle 8}$ $\,=\,$ $3^{\displaystyle 2} \times 3^{\displaystyle 3} \times 3^{\displaystyle 3x}$

$\implies$ $3^{\displaystyle 2x+8}$ $\,=\,$ $3^{\displaystyle 2+3+3x}$

$\implies$ $3^{\displaystyle 2x+8}$ $\,=\,$ $3^{\displaystyle 5+3x}$

$\implies$ $2x+8$ $\,=\,$ $5+3x$

$\implies$ $5+3x$ $\,=\,$ $2x+8$

$\implies$ $3x-2x$ $\,=\,$ $8-5$

$\,\,\,\therefore\,\,\,\,\,\,$ $x \,=\, 3$

Shortcut method

$9^{\displaystyle x+4}$ $\,=\,$ $3^{\displaystyle 2} \times 27^{\displaystyle x+1}$

$\Big(3^{\displaystyle 2}\Big)^{\displaystyle x+4}$ $\,=\,$ $3^{\displaystyle 2} \times \Big(3^{\displaystyle 3}\Big)^{\displaystyle x+1}$

$3^{\displaystyle 2 \times (x+4)}$ $\,=\,$ $3^{\displaystyle 2} \times 3^{\displaystyle 3 \times (x+1)}$

$3^{\displaystyle 2 \times x+2 \times 4}$ $\,=\,$ $3^{\displaystyle 2} \times 3^{\displaystyle 3 \times x+3 \times 1}$

$3^{\displaystyle 2x+8}$ $\,=\,$ $3^{\displaystyle 2} \times 3^{\displaystyle 3x+3}$

$3^{\displaystyle 2x+8}$ $\,=\,$ $3^{\displaystyle 2+3x+3}$

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