Algebra Examples

$z = \frac{x - μ}{\frac{σ}{\sqrt{n}}}$

Step 1

Rewrite the equation as $\frac{x - μ}{\frac{σ}{\sqrt{n}}} = z$ .

$\frac{x - μ}{\frac{σ}{\sqrt{n}}} = z$

Step 2

Multiply both sides by $\frac{σ}{\sqrt{n}}$ .

$\frac{x - μ}{\frac{σ}{\sqrt{n}}} \cdot \frac{σ}{\sqrt{n}} = z \frac{σ}{\sqrt{n}}$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{x - μ}{\frac{σ}{\sqrt{n}}} \cdot \frac{σ}{\sqrt{n}}$ .

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Step 3.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$(x - μ) \frac{\sqrt{n}}{σ} \frac{σ}{\sqrt{n}} = z \frac{σ}{\sqrt{n}}$

Step 3.1.1.2

Multiply $x - μ$ by $\frac{\sqrt{n}}{σ}$ .

$\frac{(x - μ) \sqrt{n}}{σ} \cdot \frac{σ}{\sqrt{n}} = z \frac{σ}{\sqrt{n}}$

Step 3.1.1.3

Cancel the common factor of $\sqrt{n}$ .

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Step 3.1.1.3.1

Factor $\sqrt{n}$ out of $(x - μ) \sqrt{n}$ .

$\frac{\sqrt{n} (x - μ)}{σ} \cdot \frac{σ}{\sqrt{n}} = z \frac{σ}{\sqrt{n}}$

Step 3.1.1.3.2

Cancel the common factor.

$\frac{\sqrt{n} (x - μ)}{σ} \cdot \frac{σ}{\sqrt{n}} = z \frac{σ}{\sqrt{n}}$

Step 3.1.1.3.3

Rewrite the expression.

$\frac{x - μ}{σ} σ = z \frac{σ}{\sqrt{n}}$

Step 3.1.1.4

Cancel the common factor of $σ$ .

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Step 3.1.1.4.1

Cancel the common factor.

$\frac{x - μ}{σ} σ = z \frac{σ}{\sqrt{n}}$

Step 3.1.1.4.2

Rewrite the expression.

$x - μ = z \frac{σ}{\sqrt{n}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $z \frac{σ}{\sqrt{n}}$ .

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Step 3.2.1.1

Multiply $\frac{σ}{\sqrt{n}}$ by $\frac{\sqrt{n}}{\sqrt{n}}$ .

$x - μ = z (\frac{σ}{\sqrt{n}} \cdot \frac{\sqrt{n}}{\sqrt{n}})$

Step 3.2.1.2

Combine and simplify the denominator.

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Step 3.2.1.2.1

Multiply $\frac{σ}{\sqrt{n}}$ by $\frac{\sqrt{n}}{\sqrt{n}}$ .

$x - μ = z \frac{σ \sqrt{n}}{\sqrt{n} \sqrt{n}}$

Step 3.2.1.2.2

Raise $\sqrt{n}$ to the power of $1$ .

$x - μ = z \frac{σ \sqrt{n}}{{\sqrt{n}}^{1} \sqrt{n}}$

Step 3.2.1.2.3

Raise $\sqrt{n}$ to the power of $1$ .

$x - μ = z \frac{σ \sqrt{n}}{{\sqrt{n}}^{1} {\sqrt{n}}^{1}}$

Step 3.2.1.2.4

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

$x - μ = z \frac{σ \sqrt{n}}{{\sqrt{n}}^{1 + 1}}$

Step 3.2.1.2.5

Add $1$ and $1$ .

$x - μ = z \frac{σ \sqrt{n}}{{\sqrt{n}}^{2}}$

Step 3.2.1.2.6

Rewrite ${\sqrt{n}}^{2}$ as $n$ .

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Step 3.2.1.2.6.1

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

$x - μ = z \frac{σ \sqrt{n}}{{(n^{\frac{1}{2}})}^{2}}$

Step 3.2.1.2.6.2

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

$x - μ = z \frac{σ \sqrt{n}}{n^{\frac{1}{2} \cdot 2}}$

Step 3.2.1.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$x - μ = z \frac{σ \sqrt{n}}{n^{\frac{2}{2}}}$

Step 3.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 3.2.1.2.6.4.1

Cancel the common factor.

$x - μ = z \frac{σ \sqrt{n}}{n^{\frac{2}{2}}}$

Step 3.2.1.2.6.4.2

Rewrite the expression.

$x - μ = z \frac{σ \sqrt{n}}{n^{1}}$

Step 3.2.1.2.6.5

Simplify.

$x - μ = z \frac{σ \sqrt{n}}{n}$

Step 3.2.1.3

Combine $z$ and $\frac{σ \sqrt{n}}{n}$ .

$x - μ = \frac{z σ \sqrt{n}}{n}$

Step 4

Add $μ$ to both sides of the equation.

$x = \frac{z σ \sqrt{n}}{n} + μ$