Trigonometry Examples

$x^{\frac{1}{4}} \sqrt[4]{x} = x^{a}$

Step 1

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

$x^{a} = x^{\frac{1}{4}} \sqrt[4]{x}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$x^{a} = x^{\frac{1}{4}} x^{\frac{1}{4}}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

Add $1$ and $1$ .

$x^{a} = x^{\frac{2}{4}}$

Step 3.4

Cancel the common factor of $2$ and $4$ .

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Step 3.4.1

Factor $2$ out of $2$ .

$x^{a} = x^{\frac{2 (1)}{4}}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $4$ .

$x^{a} = x^{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 3.4.2.2

Cancel the common factor.

$x^{a} = x^{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 3.4.2.3

Rewrite the expression.

$x^{a} = x^{\frac{1}{2}}$

Step 4

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

$a = \frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$a = \frac{1}{2}$

Decimal Form:

$a = 0.5$