Calculus Examples

$\frac{3^{x}}{19683} = {(3^{x})}^{4}$

Step 1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3^{x}}{19683} = 3^{x \cdot 4}$

Step 2

Raise $19683$ to the power of $1$ .

$\frac{3^{x}}{19683^{1}} = 3^{x \cdot 4}$

Step 3

Move $19683^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3^{x} \cdot 19683^{- 1} = 3^{x \cdot 4}$

Step 4

Rewrite $19683$ as $3^{9}$ .

$3^{x} \cdot {(3^{9})}^{- 1} = 3^{x \cdot 4}$

Step 5

Multiply the exponents in ${(3^{9})}^{- 1}$ .

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Step 5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3^{x} \cdot 3^{9 \cdot - 1} = 3^{x \cdot 4}$

Step 5.2

Multiply $9$ by $- 1$ .

$3^{x} \cdot 3^{- 9} = 3^{x \cdot 4}$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3^{x - 9} = 3^{x \cdot 4}$

Step 7

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x - 9 = x \cdot 4$

Step 8

Solve for $x$ .

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Step 8.1

Move $4$ to the left of $x$ .

$x - 9 = 4 x$

Step 8.2

Move all terms containing $x$ to the left side of the equation.

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Step 8.2.1

Subtract $4 x$ from both sides of the equation.

$x - 9 - 4 x = 0$

Step 8.2.2

Subtract $4 x$ from $x$ .

$- 3 x - 9 = 0$

Step 8.3

Add $9$ to both sides of the equation.

$- 3 x = 9$

Step 8.4

Divide each term in $- 3 x = 9$ by $- 3$ and simplify.

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Step 8.4.1

Divide each term in $- 3 x = 9$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{9}{- 3}$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

Cancel the common factor of $- 3$ .

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Step 8.4.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{9}{- 3}$

Step 8.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{- 3}$

Step 8.4.3

Simplify the right side.

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Step 8.4.3.1

Divide $9$ by $- 3$ .

$x = - 3$