Solve for a (x^(1/4))^5 cube root of x=x^a

${(x^{\frac{1}{4}})}^{5} \sqrt[3]{x} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = {(x^{\frac{1}{4}})}^{5} \sqrt[3]{x}$ .

$x^{a} = {(x^{\frac{1}{4}})}^{5} \sqrt[3]{x}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 3

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

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

Step 4

Multiply $x^{\frac{1}{4} \cdot 5}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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

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

Combine $\frac{1}{4}$ and $5$ .

$x^{a} = x^{\frac{5}{4} + \frac{1}{3}}$

To write $\frac{5}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{4}$ by $\frac{3}{3}$ .

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

Multiply $4$ by $3$ .

$x^{a} = x^{\frac{5 \cdot 3}{12} + \frac{1}{3} \cdot \frac{4}{4}}$

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$x^{a} = x^{\frac{5 \cdot 3}{12} + \frac{4}{3 \cdot 4}}$

Multiply $3$ by $4$ .

$x^{a} = x^{\frac{5 \cdot 3}{12} + \frac{4}{12}}$

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{5 \cdot 3 + 4}{12}}$

Simplify the numerator.

Multiply $5$ by $3$ .

$x^{a} = x^{\frac{15 + 4}{12}}$

Add $15$ and $4$ .

$x^{a} = x^{\frac{19}{12}}$

Step 5

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

$a = \frac{19}{12}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = \frac{19}{12}$

Decimal Form:

$a = 1.58\overline{3}$

Mixed Number Form:

$a = 1 \frac{7}{12}$

Solve for a (x^(1/4))^5 cube root of x=x^a

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