Precalculus Examples

$\frac{25}{\sqrt[4]{5^{x}}} = (\frac{1}{125^{x}})$

Step 1

Cross multiply.

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Step 1.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$1 \cdot (\sqrt[4]{5^{x}}) = 25 \cdot (125^{x})$

Step 1.2

Simplify the left side.

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Step 1.2.1

Multiply $\sqrt[4]{5^{x}}$ by $1$ .

$\sqrt[4]{5^{x}} = 25 \cdot (125^{x})$

Step 1.3

Simplify the right side.

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Step 1.3.1

Multiply $25 (125^{x})$ .

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Step 1.3.1.1

Rewrite $25$ as $5^{2}$ .

$\sqrt[4]{5^{x}} = 5^{2} \cdot 125^{x}$

Step 1.3.1.2

Rewrite $125$ as $5^{3}$ .

$\sqrt[4]{5^{x}} = 5^{2} \cdot {(5^{3})}^{x}$

Step 1.3.1.3

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

$\sqrt[4]{5^{x}} = 5^{2} \cdot 5^{3 x}$

Step 1.3.1.4

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

$\sqrt[4]{5^{x}} = 5^{2 + 3 x}$

Step 2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{5^{x}}}^{4} = {(5^{2 + 3 x})}^{4}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{5^{x}}$ as $5^{\frac{x}{4}}$ .

${(5^{\frac{x}{4}})}^{4} = {(5^{2 + 3 x})}^{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply the exponents in ${(5^{\frac{x}{4}})}^{4}$ .

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Step 3.2.1.1

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

$5^{\frac{x}{4} \cdot 4} = {(5^{2 + 3 x})}^{4}$

Step 3.2.1.2

Cancel the common factor of $4$ .

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Step 3.2.1.2.1

Cancel the common factor.

$5^{\frac{x}{4} \cdot 4} = {(5^{2 + 3 x})}^{4}$

Step 3.2.1.2.2

Rewrite the expression.

$5^{x} = {(5^{2 + 3 x})}^{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the exponents in ${(5^{2 + 3 x})}^{4}$ .

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Step 3.3.1.1

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

$5^{x} = 5^{(2 + 3 x) \cdot 4}$

Step 3.3.1.2

Apply the distributive property.

$5^{x} = 5^{2 \cdot 4 + 3 x \cdot 4}$

Step 3.3.1.3

Multiply $2$ by $4$ .

$5^{x} = 5^{8 + 3 x \cdot 4}$

Step 3.3.1.4

Multiply $4$ by $3$ .

$5^{x} = 5^{8 + 12 x}$

Step 4

Solve for $x$ .

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Step 4.1

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

$x = 8 + 12 x$

Step 4.2

Solve for $x$ .

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Step 4.2.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.2.1.1

Subtract $12 x$ from both sides of the equation.

$x - 12 x = 8$

Step 4.2.1.2

Subtract $12 x$ from $x$ .

$- 11 x = 8$

Step 4.2.2

Divide each term in $- 11 x = 8$ by $- 11$ and simplify.

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Step 4.2.2.1

Divide each term in $- 11 x = 8$ by $- 11$ .

$\frac{- 11 x}{- 11} = \frac{8}{- 11}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $- 11$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{- 11 x}{- 11} = \frac{8}{- 11}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{- 11}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{8}{11}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{8}{11}$

Decimal Form:

$x = - 0.\overline{72}$