Solve for x z=(x-mu)/(sigma/( square root of n))

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

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.

Simplify the left side.

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $σ$ .

Cancel the common factor.

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

Rewrite the expression.

$x - μ = z \frac{σ}{\sqrt{n}}$

Simplify the right side.

Simplify $z \frac{σ}{\sqrt{n}}$ .

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x - μ = z \frac{σ \sqrt{n}}{n^{1}}$

Simplify.

$x - μ = z \frac{σ \sqrt{n}}{n}$

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.

$x = \frac{z σ \sqrt{n}}{n} + μ$

Solve for x z=(x-mu)/(sigma/( square root of n))

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