Calculus Examples

$243^{2 k} (9^{2 k - 1}) = 9$

Step 1

Rewrite $243$ as $3^{5}$ .

${(3^{5})}^{2 k} \cdot 9^{2 k - 1} = 9$

Step 2

Multiply the exponents in ${(3^{5})}^{2 k}$ .

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Step 2.1

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

$3^{5 (2 k)} \cdot 9^{2 k - 1} = 9$

Step 2.2

Multiply $2$ by $5$ .

$3^{10 k} \cdot 9^{2 k - 1} = 9$

Step 3

Rewrite $9$ as $3^{2}$ .

$3^{10 k} \cdot {(3^{2})}^{2 k - 1} = 9$

Step 4

Multiply the exponents in ${(3^{2})}^{2 k - 1}$ .

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Step 4.1

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

$3^{10 k} \cdot 3^{2 (2 k - 1)} = 9$

Step 4.2

Apply the distributive property.

$3^{10 k} \cdot 3^{2 (2 k) + 2 \cdot - 1} = 9$

Step 4.3

Multiply $2$ by $2$ .

$3^{10 k} \cdot 3^{4 k + 2 \cdot - 1} = 9$

Step 4.4

Multiply $2$ by $- 1$ .

$3^{10 k} \cdot 3^{4 k - 2} = 9$

Step 5

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

$3^{10 k + 4 k - 2} = 9$

Step 6

Add $10 k$ and $4 k$ .

$3^{14 k - 2} = 9$

Step 7

Create equivalent expressions in the equation that all have equal bases.

$3^{14 k - 2} = 3^{2}$

Step 8

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

$14 k - 2 = 2$

Step 9

Solve for $k$ .

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Step 9.1

Move all terms not containing $k$ to the right side of the equation.

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Step 9.1.1

Add $2$ to both sides of the equation.

$14 k = 2 + 2$

Step 9.1.2

Add $2$ and $2$ .

$14 k = 4$

Step 9.2

Divide each term in $14 k = 4$ by $14$ and simplify.

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Step 9.2.1

Divide each term in $14 k = 4$ by $14$ .

$\frac{14 k}{14} = \frac{4}{14}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $14$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{14 k}{14} = \frac{4}{14}$

Step 9.2.2.1.2

Divide $k$ by $1$ .

$k = \frac{4}{14}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Cancel the common factor of $4$ and $14$ .

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Step 9.2.3.1.1

Factor $2$ out of $4$ .

$k = \frac{2 (2)}{14}$

Step 9.2.3.1.2

Cancel the common factors.

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Step 9.2.3.1.2.1

Factor $2$ out of $14$ .

$k = \frac{2 \cdot 2}{2 \cdot 7}$

Step 9.2.3.1.2.2

Cancel the common factor.

$k = \frac{2 \cdot 2}{2 \cdot 7}$

Step 9.2.3.1.2.3

Rewrite the expression.

$k = \frac{2}{7}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$k = \frac{2}{7}$

Decimal Form:

$k = 0.\overline{285714}$