Precalculus Examples

$3 x^{- \frac{1}{2}} = 27$

Step 1

Divide each term in $3 x^{- \frac{1}{2}} = 27$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 x^{- \frac{1}{2}} = 27$ by $3$ .

$\frac{3 x^{- \frac{1}{2}}}{3} = \frac{27}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{3 x^{\frac{1}{2}}} = \frac{27}{3}$

Step 1.2.2

Cancel the common factor.

$\frac{3}{3 x^{\frac{1}{2}}} = \frac{27}{3}$

Step 1.2.3

Rewrite the expression.

$\frac{1}{x^{\frac{1}{2}}} = \frac{27}{3}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $27$ by $3$ .

$\frac{1}{x^{\frac{1}{2}}} = 9$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{\frac{1}{2}}, 1$

Step 2.2

The LCM of one and any expression is the expression.

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

Step 3

Multiply each term in $\frac{1}{x^{\frac{1}{2}}} = 9$ by $x^{\frac{1}{2}}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{1}{x^{\frac{1}{2}}} = 9$ by $x^{\frac{1}{2}}$ .

$\frac{1}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 9 x^{\frac{1}{2}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{1}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 9 x^{\frac{1}{2}}$

Step 3.2.1.2

Rewrite the expression.

$1 = 9 x^{\frac{1}{2}}$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $9 x^{\frac{1}{2}} = 1$ .

$9 x^{\frac{1}{2}} = 1$

Step 4.2

Divide each term in $9 x^{\frac{1}{2}} = 1$ by $9$ and simplify.

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Step 4.2.1

Divide each term in $9 x^{\frac{1}{2}} = 1$ by $9$ .

$\frac{9 x^{\frac{1}{2}}}{9} = \frac{1}{9}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor.

$\frac{9 x^{\frac{1}{2}}}{9} = \frac{1}{9}$

Step 4.2.2.2

Divide $x^{\frac{1}{2}}$ by $1$ .

$x^{\frac{1}{2}} = \frac{1}{9}$

Step 4.3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = {(\frac{1}{9})}^{2}$

Step 4.4

Simplify the exponent.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.4.1.1.1

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

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Step 4.4.1.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = {(\frac{1}{9})}^{2}$

Step 4.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 4.4.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\frac{1}{9})}^{2}$

Step 4.4.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(\frac{1}{9})}^{2}$

Step 4.4.1.1.2

Simplify.

$x = {(\frac{1}{9})}^{2}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify ${(\frac{1}{9})}^{2}$ .

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Step 4.4.2.1.1

Apply the product rule to $\frac{1}{9}$ .

$x = \frac{1^{2}}{9^{2}}$

Step 4.4.2.1.2

One to any power is one.

$x = \frac{1}{9^{2}}$

Step 4.4.2.1.3

Raise $9$ to the power of $2$ .

$x = \frac{1}{81}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{81}$

Decimal Form:

$x = 0.01234567 \dots$