Solve for x 81 fifth root of 27=3^x

$81 \sqrt[5]{27} = 3^{x}$

Step 1

Rewrite the equation as $3^{x} = 81 \sqrt[5]{27}$ .

$3^{x} = 81 \sqrt[5]{27}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{27}$ as $27^{\frac{1}{5}}$ .

$3^{x} = 81 \cdot 27^{\frac{1}{5}}$

Step 3

Rewrite $81$ as $3^{4}$ .

$3^{x} = 3^{4} \cdot 27^{\frac{1}{5}}$

Step 4

Rewrite $27$ as $3^{3}$ .

$3^{x} = 3^{4} \cdot {(3^{3})}^{\frac{1}{5}}$

Step 5

Multiply the exponents in ${(3^{3})}^{\frac{1}{5}}$ .

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

$3^{x} = 3^{4} \cdot 3^{3 (\frac{1}{5})}$

Combine $3$ and $\frac{1}{5}$ .

$3^{x} = 3^{4} \cdot 3^{\frac{3}{5}}$

Step 6

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

$3^{x} = 3^{4 + \frac{3}{5}}$

Step 7

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$3^{x} = 3^{4 \cdot \frac{5}{5} + \frac{3}{5}}$

Step 8

Combine $4$ and $\frac{5}{5}$ .

$3^{x} = 3^{\frac{4 \cdot 5}{5} + \frac{3}{5}}$

Step 9

Combine the numerators over the common denominator.

$3^{x} = 3^{\frac{4 \cdot 5 + 3}{5}}$

Step 10

Simplify the numerator.

Multiply $4$ by $5$ .

$3^{x} = 3^{\frac{20 + 3}{5}}$

Add $20$ and $3$ .

$3^{x} = 3^{\frac{23}{5}}$

Step 11

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

$x = \frac{23}{5}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = \frac{23}{5}$

Decimal Form:

$x = 4.6$

Mixed Number Form:

$x = 4 \frac{3}{5}$

Solve for x 81 fifth root of 27=3^x

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