Precalculus Examples

$\log (3) \sqrt{\frac{7^{3}}{5^{4}}}$

Step 1

Raise $7$ to the power of $3$ .

$\log (3) \sqrt{\frac{343}{5^{4}}}$

Step 2

Raise $5$ to the power of $4$ .

$\log (3) \sqrt{\frac{343}{625}}$

Step 3

Rewrite $\sqrt{\frac{343}{625}}$ as $\frac{\sqrt{343}}{\sqrt{625}}$ .

$\log (3) \frac{\sqrt{343}}{\sqrt{625}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $343$ as $7^{2} \cdot 7$ .

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Step 4.1.1

Factor $49$ out of $343$ .

$\log (3) \frac{\sqrt{49 (7)}}{\sqrt{625}}$

Step 4.1.2

Rewrite $49$ as $7^{2}$ .

$\log (3) \frac{\sqrt{7^{2} \cdot 7}}{\sqrt{625}}$

Step 4.2

Pull terms out from under the radical.

$\log (3) \frac{7 \sqrt{7}}{\sqrt{625}}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $625$ as $25^{2}$ .

$\log (3) \frac{7 \sqrt{7}}{\sqrt{25^{2}}}$

Step 5.2

Pull terms out from under the radical, assuming positive real numbers.

$\log (3) \frac{7 \sqrt{7}}{25}$

Step 6

Combine $\log (3)$ and $\frac{7 \sqrt{7}}{25}$ .

$\frac{\log (3) (7 \sqrt{7})}{25}$

Step 7

Move $7$ to the left of $\log (3)$ .

$\frac{7 \log (3) \sqrt{7}}{25}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{7 \log (3) \sqrt{7}}{25}$

Decimal Form:

$0.35345637 \dots$