Basic Math Examples

$11.8 + z^{7.8} = 4.5$

Step 1

Move all terms not containing $z$ to the right side of the equation.

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Step 1.1

Subtract $11.8$ from both sides of the equation.

$z^{7.8} = 4.5 - 11.8$

Step 1.2

Subtract $11.8$ from $4.5$ .

$z^{7.8} = - 7.3$

Step 2

Convert the decimal exponent to a fractional exponent.

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Step 2.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there is $1$ number to the right of the decimal point, place the decimal number over $10^{1}$ $(10)$ . Next, add the whole number to the left of the decimal.

$z^{7 \frac{8}{10}} = - 7.3$

Step 2.2

Reduce the fractional part of the mixed number.

$z^{7 \frac{4}{5}} = - 7.3$

Step 2.3

Convert $7 \frac{4}{5}$ to an improper fraction.

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Step 2.3.1

A mixed number is an addition of its whole and fractional parts.

$z^{7 + \frac{4}{5}} = - 7.3$

Step 2.3.2

Add $7$ and $\frac{4}{5}$ .

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Step 2.3.2.1

To write $7$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$z^{7 \cdot \frac{5}{5} + \frac{4}{5}} = - 7.3$

Step 2.3.2.2

Combine $7$ and $\frac{5}{5}$ .

$z^{\frac{7 \cdot 5}{5} + \frac{4}{5}} = - 7.3$

Step 2.3.2.3

Combine the numerators over the common denominator.

$z^{\frac{7 \cdot 5 + 4}{5}} = - 7.3$

Step 2.3.2.4

Simplify the numerator.

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Step 2.3.2.4.1

Multiply $7$ by $5$ .

$z^{\frac{35 + 4}{5}} = - 7.3$

Step 2.3.2.4.2

Add $35$ and $4$ .

$z^{\frac{39}{5}} = - 7.3$

Step 3

Raise each side of the equation to the power of $\frac{1}{7.8}$ to eliminate the fractional exponent on the left side.

${(z^{\frac{39}{5}})}^{\frac{1}{7.8}} = {(- 7.3)}^{\frac{1}{7.8}}$

Step 4

Simplify the exponent.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify ${(z^{\frac{39}{5}})}^{\frac{1}{7.8}}$ .

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Step 4.1.1.1

Multiply the exponents in ${(z^{\frac{39}{5}})}^{\frac{1}{7.8}}$ .

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Step 4.1.1.1.1

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

$z^{\frac{39}{5} \cdot \frac{1}{7.8}} = {(- 7.3)}^{\frac{1}{7.8}}$

Step 4.1.1.1.2

Cancel the common factor of $7.8$ .

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Step 4.1.1.1.2.1

Factor $7.8$ out of $39$ .

$z^{\frac{7.8 (5)}{5} \cdot \frac{1}{7.8}} = {(- 7.3)}^{\frac{1}{7.8}}$

Step 4.1.1.1.2.2

Cancel the common factor.

$z^{\frac{7.8 \cdot 5}{5} \cdot \frac{1}{7.8}} = {(- 7.3)}^{\frac{1}{7.8}}$

Step 4.1.1.1.2.3

Rewrite the expression.

$z^{\frac{5}{5}} = {(- 7.3)}^{\frac{1}{7.8}}$

Step 4.1.1.1.3

Divide $5$ by $5$ .

$z^{1} = {(- 7.3)}^{\frac{1}{7.8}}$

Step 4.1.1.2

Simplify.

$z = {(- 7.3)}^{\frac{1}{7.8}}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Divide $1$ by $7.8$ .

$z = {(- 7.3)}^{0.\overline{128205}}$

Step 5

Exclude the solutions that do not make $11.8 + z^{7.8} = 4.5$ true.

$No solution$