Precalculus Examples

$49^{x - 1} = 7 \sqrt{7}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$49^{x - 1} = 7 \cdot 7^{\frac{1}{2}}$

Step 2

Multiply $7$ by $7^{\frac{1}{2}}$ by adding the exponents.

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Step 2.1

Multiply $7$ by $7^{\frac{1}{2}}$ .

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Step 2.1.1

Raise $7$ to the power of $1$ .

$49^{x - 1} = 7^{1} \cdot 7^{\frac{1}{2}}$

Step 2.1.2

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

$49^{x - 1} = 7^{1 + \frac{1}{2}}$

Step 2.2

Write $1$ as a fraction with a common denominator.

$49^{x - 1} = 7^{\frac{2}{2} + \frac{1}{2}}$

Step 2.3

Combine the numerators over the common denominator.

$49^{x - 1} = 7^{\frac{2 + 1}{2}}$

Step 2.4

Add $2$ and $1$ .

$49^{x - 1} = 7^{\frac{3}{2}}$

Step 3

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

$7^{2 (x - 1)} = 7^{\frac{3}{2}}$

Step 4

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

$2 (x - 1) = \frac{3}{2}$

Step 5

Solve for $x$ .

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Step 5.1

Divide each term in $2 (x - 1) = \frac{3}{2}$ by $2$ and simplify.

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Step 5.1.1

Divide each term in $2 (x - 1) = \frac{3}{2}$ by $2$ .

$\frac{2 (x - 1)}{2} = \frac{\frac{3}{2}}{2}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $2$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{2 (x - 1)}{2} = \frac{\frac{3}{2}}{2}$

Step 5.1.2.1.2

Divide $x - 1$ by $1$ .

$x - 1 = \frac{\frac{3}{2}}{2}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Multiply the numerator by the reciprocal of the denominator.

$x - 1 = \frac{3}{2} \cdot \frac{1}{2}$

Step 5.1.3.2

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

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Step 5.1.3.2.1

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$x - 1 = \frac{3}{2 \cdot 2}$

Step 5.1.3.2.2

Multiply $2$ by $2$ .

$x - 1 = \frac{3}{4}$

Step 5.2

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

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Step 5.2.1

Add $1$ to both sides of the equation.

$x = \frac{3}{4} + 1$

Step 5.2.2

Write $1$ as a fraction with a common denominator.

$x = \frac{3}{4} + \frac{4}{4}$

Step 5.2.3

Combine the numerators over the common denominator.

$x = \frac{3 + 4}{4}$

Step 5.2.4

Add $3$ and $4$ .

$x = \frac{7}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{4}$

Decimal Form:

$x = 1.75$

Mixed Number Form:

$x = 1 \frac{3}{4}$