Linear Algebra Examples

$3^{2 x - 7} = \sqrt{81^{x}}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{81^{x}} = 3^{2 x - 7}$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{81^{x}}}^{2} = {(3^{2 x - 7})}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{81^{x}}$ as $81^{\frac{x}{2}}$ .

${(81^{\frac{x}{2}})}^{2} = {(3^{2 x - 7})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply the exponents in ${(81^{\frac{x}{2}})}^{2}$ .

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Step 3.2.1.1

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

$81^{\frac{x}{2} \cdot 2} = {(3^{2 x - 7})}^{2}$

Step 3.2.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.2.1

Cancel the common factor.

$81^{\frac{x}{2} \cdot 2} = {(3^{2 x - 7})}^{2}$

Step 3.2.1.2.2

Rewrite the expression.

$81^{x} = {(3^{2 x - 7})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the exponents in ${(3^{2 x - 7})}^{2}$ .

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Step 3.3.1.1

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

$81^{x} = 3^{(2 x - 7) \cdot 2}$

Step 3.3.1.2

Apply the distributive property.

$81^{x} = 3^{2 x \cdot 2 - 7 \cdot 2}$

Step 3.3.1.3

Multiply $2$ by $2$ .

$81^{x} = 3^{4 x - 7 \cdot 2}$

Step 3.3.1.4

Multiply $- 7$ by $2$ .

$81^{x} = 3^{4 x - 14}$

Step 4

Solve for $x$ .

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Step 4.1

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

$3^{4 x} = 3^{4 x - 14}$

Step 4.2

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

$4 x = 4 x - 14$

Step 4.3

Solve for $x$ .

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Step 4.3.1

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

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Step 4.3.1.1

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

$4 x - 4 x = - 14$

Step 4.3.1.2

Subtract $4 x$ from $4 x$ .

$0 = - 14$

Step 4.3.2

Since $0 \neq - 14$ , there are no solutions.

No solution