Finite Math Examples

$x^{\frac{1}{2}} + {(\sqrt{3 x})}^{\frac{1}{4}} - 18 = 0$

Step 1

Simplify the left side.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

$x^{\frac{1}{2}} + {({(3 x)}^{\frac{1}{2}})}^{\frac{1}{4}} - 18 = 0$

Step 1.2

Multiply the exponents in ${({(3 x)}^{\frac{1}{2}})}^{\frac{1}{4}}$ .

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Step 1.2.1

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

$x^{\frac{1}{2}} + {(3 x)}^{\frac{1}{2} \cdot \frac{1}{4}} - 18 = 0$

Step 1.2.2

Multiply $\frac{1}{2} \cdot \frac{1}{4}$ .

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Step 1.2.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$x^{\frac{1}{2}} + {(3 x)}^{\frac{1}{2 \cdot 4}} - 18 = 0$

Step 1.2.2.2

Multiply $2$ by $4$ .

$x^{\frac{1}{2}} + {(3 x)}^{\frac{1}{8}} - 18 = 0$

Step 2

Find a common factor $3^{\frac{1}{8}} x^{\frac{1}{8}}$ that is present in each term.

${(x^{\frac{1}{8} + \frac{1}{8}})}^{2} {({(3 x)}^{\frac{1}{8} + \frac{1}{8}})}^{\frac{1}{2}} - 18$

Step 3

Substitute $u$ for $3^{\frac{1}{8}} x^{\frac{1}{8}}$ .

${(x^{\frac{1}{8} + \frac{1}{8}})}^{2} {({(3 x)}^{\frac{1}{8} + \frac{1}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4

Solve for $u$ .

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Step 4.1

Simplify each term.

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Step 4.1.1

Combine the numerators over the common denominator.

${(x^{\frac{1 + 1}{8}})}^{2} {({(3 x)}^{\frac{1}{8} + \frac{1}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4.1.2

Add $1$ and $1$ .

${(x^{\frac{2}{8}})}^{2} {({(3 x)}^{\frac{1}{8} + \frac{1}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4.1.3

Multiply the exponents in ${(x^{\frac{2}{8}})}^{2}$ .

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Step 4.1.3.1

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

$x^{\frac{2}{8} \cdot 2} {({(3 x)}^{\frac{1}{8} + \frac{1}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4.1.3.2

Cancel the common factor of $2$ .

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Step 4.1.3.2.1

Factor $2$ out of $8$ .

$x^{\frac{2}{2 (4)} \cdot 2} {({(3 x)}^{\frac{1}{8} + \frac{1}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4.1.3.2.2

Cancel the common factor.

$x^{\frac{2}{2 \cdot 4} \cdot 2} {({(3 x)}^{\frac{1}{8} + \frac{1}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4.1.3.2.3

Rewrite the expression.

$x^{\frac{2}{4}} {({(3 x)}^{\frac{1}{8} + \frac{1}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4.1.3.3

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

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Step 4.1.3.3.1

Factor $2$ out of $2$ .

$x^{\frac{2 (1)}{4}} {({(3 x)}^{\frac{1}{8} + \frac{1}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4.1.3.3.2

Cancel the common factors.

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Step 4.1.3.3.2.1

Factor $2$ out of $4$ .

$x^{\frac{2 \cdot 1}{2 \cdot 2}} {({(3 x)}^{\frac{1}{8} + \frac{1}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4.1.3.3.2.2

Cancel the common factor.

$x^{\frac{2 \cdot 1}{2 \cdot 2}} {({(3 x)}^{\frac{1}{8} + \frac{1}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4.1.3.3.2.3

Rewrite the expression.

$x^{\frac{1}{2}} {({(3 x)}^{\frac{1}{8} + \frac{1}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4.1.4

Combine the numerators over the common denominator.

$x^{\frac{1}{2}} {({(3 x)}^{\frac{1 + 1}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4.1.5

Add $1$ and $1$ .

$x^{\frac{1}{2}} {({(3 x)}^{\frac{2}{8}})}^{\frac{1}{2}} - 18 = 0$

Step 4.1.6

Multiply the exponents in ${({(3 x)}^{\frac{2}{8}})}^{\frac{1}{2}}$ .

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Step 4.1.6.1

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

$x^{\frac{1}{2}} {(3 x)}^{\frac{2}{8} \cdot \frac{1}{2}} - 18 = 0$

Step 4.1.6.2

Cancel the common factor of $2$ .

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Step 4.1.6.2.1

Cancel the common factor.

$x^{\frac{1}{2}} {(3 x)}^{\frac{2}{8} \cdot \frac{1}{2}} - 18 = 0$

Step 4.1.6.2.2

Rewrite the expression.

$x^{\frac{1}{2}} {(3 x)}^{\frac{1}{8}} - 18 = 0$

Step 4.1.7

Apply the product rule to $3 x$ .

$x^{\frac{1}{2}} (3^{\frac{1}{8}} x^{\frac{1}{8}}) - 18 = 0$

Step 4.1.8

Rewrite using the commutative property of multiplication.

$3^{\frac{1}{8}} x^{\frac{1}{2}} x^{\frac{1}{8}} - 18 = 0$

Step 4.1.9

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{8}}$ by adding the exponents.

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Step 4.1.9.1

Move $x^{\frac{1}{8}}$ .

$3^{\frac{1}{8}} (x^{\frac{1}{8}} x^{\frac{1}{2}}) - 18 = 0$

Step 4.1.9.2

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

$3^{\frac{1}{8}} x^{\frac{1}{8} + \frac{1}{2}} - 18 = 0$

Step 4.1.9.3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$3^{\frac{1}{8}} x^{\frac{1}{8} + \frac{1}{2} \cdot \frac{4}{4}} - 18 = 0$

Step 4.1.9.4

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.9.4.1

Multiply $\frac{1}{2}$ by $\frac{4}{4}$ .

$3^{\frac{1}{8}} x^{\frac{1}{8} + \frac{4}{2 \cdot 4}} - 18 = 0$

Step 4.1.9.4.2

Multiply $2$ by $4$ .

$3^{\frac{1}{8}} x^{\frac{1}{8} + \frac{4}{8}} - 18 = 0$

Step 4.1.9.5

Combine the numerators over the common denominator.

$3^{\frac{1}{8}} x^{\frac{1 + 4}{8}} - 18 = 0$

Step 4.1.9.6

Add $1$ and $4$ .

$3^{\frac{1}{8}} x^{\frac{5}{8}} - 18 = 0$

Step 4.2

Add $18$ to both sides of the equation.

$3^{\frac{1}{8}} x^{\frac{5}{8}} = 18$

Step 4.3

Raise each side of the equation to the power of $\frac{8}{5}$ to eliminate the fractional exponent on the left side.

${(3^{\frac{1}{8}} x^{\frac{5}{8}})}^{\frac{8}{5}} = 18^{\frac{8}{5}}$

Step 4.4

Simplify the left side.

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Step 4.4.1

Simplify ${(3^{\frac{1}{8}} x^{\frac{5}{8}})}^{\frac{8}{5}}$ .

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Step 4.4.1.1

Apply the product rule to $3^{\frac{1}{8}} x^{\frac{5}{8}}$ .

${(3^{\frac{1}{8}})}^{\frac{8}{5}} {(x^{\frac{5}{8}})}^{\frac{8}{5}} = 18^{\frac{8}{5}}$

Step 4.4.1.2

Multiply the exponents in ${(3^{\frac{1}{8}})}^{\frac{8}{5}}$ .

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Step 4.4.1.2.1

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

$3^{\frac{1}{8} \cdot \frac{8}{5}} {(x^{\frac{5}{8}})}^{\frac{8}{5}} = 18^{\frac{8}{5}}$

Step 4.4.1.2.2

Cancel the common factor of $8$ .

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Step 4.4.1.2.2.1

Cancel the common factor.

$3^{\frac{1}{8} \cdot \frac{8}{5}} {(x^{\frac{5}{8}})}^{\frac{8}{5}} = 18^{\frac{8}{5}}$

Step 4.4.1.2.2.2

Rewrite the expression.

$3^{\frac{1}{5}} {(x^{\frac{5}{8}})}^{\frac{8}{5}} = 18^{\frac{8}{5}}$

Step 4.4.1.3

Multiply the exponents in ${(x^{\frac{5}{8}})}^{\frac{8}{5}}$ .

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Step 4.4.1.3.1

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

$3^{\frac{1}{5}} x^{\frac{5}{8} \cdot \frac{8}{5}} = 18^{\frac{8}{5}}$

Step 4.4.1.3.2

Cancel the common factor of $5$ .

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Step 4.4.1.3.2.1

Cancel the common factor.

$3^{\frac{1}{5}} x^{\frac{5}{8} \cdot \frac{8}{5}} = 18^{\frac{8}{5}}$

Step 4.4.1.3.2.2

Rewrite the expression.

$3^{\frac{1}{5}} x^{\frac{1}{8} \cdot 8} = 18^{\frac{8}{5}}$

Step 4.4.1.3.3

Cancel the common factor of $8$ .

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Step 4.4.1.3.3.1

Cancel the common factor.

$3^{\frac{1}{5}} x^{\frac{1}{8} \cdot 8} = 18^{\frac{8}{5}}$

Step 4.4.1.3.3.2

Rewrite the expression.

$3^{\frac{1}{5}} x^{1} = 18^{\frac{8}{5}}$

Step 4.4.1.4

Simplify.

$3^{\frac{1}{5}} x = 18^{\frac{8}{5}}$

Step 4.4.1.5

Reorder factors in $3^{\frac{1}{5}} x$ .

$x \cdot 3^{\frac{1}{5}} = 18^{\frac{8}{5}}$

Step 4.5

Divide each term in $x \cdot 3^{\frac{1}{5}} = 18^{\frac{8}{5}}$ by $3^{\frac{1}{5}}$ and simplify.

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Step 4.5.1

Divide each term in $x \cdot 3^{\frac{1}{5}} = 18^{\frac{8}{5}}$ by $3^{\frac{1}{5}}$ .

$\frac{x \cdot 3^{\frac{1}{5}}}{3^{\frac{1}{5}}} = \frac{18^{\frac{8}{5}}}{3^{\frac{1}{5}}}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor.

$\frac{x \cdot 3^{\frac{1}{5}}}{3^{\frac{1}{5}}} = \frac{18^{\frac{8}{5}}}{3^{\frac{1}{5}}}$

Step 4.5.2.2

Divide $x$ by $1$ .

$x = \frac{18^{\frac{8}{5}}}{3^{\frac{1}{5}}}$

Step 5

Exclude the solutions that do not make $x^{\frac{1}{2}} + {(\sqrt{3 x})}^{\frac{1}{4}} - 18 = 0$ true.

$No solution$