Solve for a (x^3)^(1/5) sixth root of x^5=x^a

${(x^{3})}^{\frac{1}{5}} \sqrt[6]{x^{5}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = {(x^{3})}^{\frac{1}{5}} \sqrt[6]{x^{5}}$ .

$x^{a} = {(x^{3})}^{\frac{1}{5}} \sqrt[6]{x^{5}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x^{5}}$ as $x^{\frac{5}{6}}$ .

$x^{a} = {(x^{3})}^{\frac{1}{5}} x^{\frac{5}{6}}$

Step 3

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

$x^{a} = x^{3 (\frac{1}{5})} x^{\frac{5}{6}}$

Step 4

Multiply $x^{3 (\frac{1}{5})}$ by $x^{\frac{5}{6}}$ by adding the exponents.

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

$x^{a} = x^{3 (\frac{1}{5}) + \frac{5}{6}}$

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

$x^{a} = x^{\frac{3}{5} + \frac{5}{6}}$

To write $\frac{3}{5}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x^{a} = x^{\frac{3}{5} \cdot \frac{6}{6} + \frac{5}{6}}$

To write $\frac{5}{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{a} = x^{\frac{3}{5} \cdot \frac{6}{6} + \frac{5}{6} \cdot \frac{5}{5}}$

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{5}$ by $\frac{6}{6}$ .

$x^{a} = x^{\frac{3 \cdot 6}{5 \cdot 6} + \frac{5}{6} \cdot \frac{5}{5}}$

Multiply $5$ by $6$ .

$x^{a} = x^{\frac{3 \cdot 6}{30} + \frac{5}{6} \cdot \frac{5}{5}}$

Multiply $\frac{5}{6}$ by $\frac{5}{5}$ .

$x^{a} = x^{\frac{3 \cdot 6}{30} + \frac{5 \cdot 5}{6 \cdot 5}}$

Multiply $6$ by $5$ .

$x^{a} = x^{\frac{3 \cdot 6}{30} + \frac{5 \cdot 5}{30}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{3 \cdot 6 + 5 \cdot 5}{30}}$

Simplify the numerator.

Multiply $3$ by $6$ .

$x^{a} = x^{\frac{18 + 5 \cdot 5}{30}}$

Multiply $5$ by $5$ .

$x^{a} = x^{\frac{18 + 25}{30}}$

Add $18$ and $25$ .

$x^{a} = x^{\frac{43}{30}}$

Step 5

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

$a = \frac{43}{30}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = \frac{43}{30}$

Decimal Form:

$a = 1.4\overline{3}$

Mixed Number Form:

$a = 1 \frac{13}{30}$

Solve for a (x^3)^(1/5) sixth root of x^5=x^a

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