Algebra Examples

$2 a + a c^{2} - a^{2} c - 2 c$ for $a = 1 \frac{1}{3}$ and $c = - 1 \frac{2}{3}$

Step 1

The system is already solved.

$a = 1 \frac{1}{3}, c = - \frac{5}{3}$

Step 2

Replace the variable $a$ with $1 \frac{1}{3}$ in the expression.

$2 (1 \frac{1}{3}) + (1 \frac{1}{3}) \cdot c^{2} - {(1 \frac{1}{3})}^{2} c - 2 c$

Step 3

Replace the variable $c$ with $- \frac{5}{3}$ in the expression.

$2 (1 \frac{1}{3}) + (1 \frac{1}{3}) \cdot {(- \frac{5}{3})}^{2} - {(1 \frac{1}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 4

Convert $1 \frac{1}{3}$ to an improper fraction.

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Step 4.1

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

$2 (1 + \frac{1}{3}) + (1 \frac{1}{3}) \cdot {(- \frac{5}{3})}^{2} - {(1 \frac{1}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 4.2

Add $1$ and $\frac{1}{3}$ .

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Step 4.2.1

Write $1$ as a fraction with a common denominator.

$2 (\frac{3}{3} + \frac{1}{3}) + (1 \frac{1}{3}) \cdot {(- \frac{5}{3})}^{2} - {(1 \frac{1}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 4.2.2

Combine the numerators over the common denominator.

$2 \frac{3 + 1}{3} + (1 \frac{1}{3}) \cdot {(- \frac{5}{3})}^{2} - {(1 \frac{1}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 4.2.3

Add $3$ and $1$ .

$2 (\frac{4}{3}) + (1 \frac{1}{3}) \cdot {(- \frac{5}{3})}^{2} - {(1 \frac{1}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 5

Convert $1 \frac{1}{3}$ to an improper fraction.

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Step 5.1

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

$2 (\frac{4}{3}) + (1 + \frac{1}{3}) \cdot {(- \frac{5}{3})}^{2} - {(1 \frac{1}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 5.2

Add $1$ and $\frac{1}{3}$ .

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Step 5.2.1

Write $1$ as a fraction with a common denominator.

$2 (\frac{4}{3}) + (\frac{3}{3} + \frac{1}{3}) \cdot {(- \frac{5}{3})}^{2} - {(1 \frac{1}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 5.2.2

Combine the numerators over the common denominator.

$2 (\frac{4}{3}) + \frac{3 + 1}{3} \cdot {(- \frac{5}{3})}^{2} - {(1 \frac{1}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 5.2.3

Add $3$ and $1$ .

$2 (\frac{4}{3}) + \frac{4}{3} \cdot {(- \frac{5}{3})}^{2} - {(1 \frac{1}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 6

Convert $1 \frac{1}{3}$ to an improper fraction.

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Step 6.1

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

$2 (\frac{4}{3}) + \frac{4}{3} \cdot {(- \frac{5}{3})}^{2} - {(1 + \frac{1}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 6.2

Add $1$ and $\frac{1}{3}$ .

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Step 6.2.1

Write $1$ as a fraction with a common denominator.

$2 (\frac{4}{3}) + \frac{4}{3} \cdot {(- \frac{5}{3})}^{2} - {(\frac{3}{3} + \frac{1}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 6.2.2

Combine the numerators over the common denominator.

$2 (\frac{4}{3}) + \frac{4}{3} \cdot {(- \frac{5}{3})}^{2} - {(\frac{3 + 1}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 6.2.3

Add $3$ and $1$ .

$2 (\frac{4}{3}) + \frac{4}{3} \cdot {(- \frac{5}{3})}^{2} - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7

Simplify each term.

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Step 7.1

Multiply $2 (\frac{4}{3})$ .

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Step 7.1.1

Combine $2$ and $\frac{4}{3}$ .

$\frac{2 \cdot 4}{3} + \frac{4}{3} \cdot {(- \frac{5}{3})}^{2} - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.1.2

Multiply $2$ by $4$ .

$\frac{8}{3} + \frac{4}{3} \cdot {(- \frac{5}{3})}^{2} - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.2.1

Apply the product rule to $- \frac{5}{3}$ .

$\frac{8}{3} + \frac{4}{3} \cdot ({(- 1)}^{2} {(\frac{5}{3})}^{2}) - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.2.2

Apply the product rule to $\frac{5}{3}$ .

$\frac{8}{3} + \frac{4}{3} \cdot ({(- 1)}^{2} \frac{5^{2}}{3^{2}}) - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.3

Raise $- 1$ to the power of $2$ .

$\frac{8}{3} + \frac{4}{3} \cdot (1 \frac{5^{2}}{3^{2}}) - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.4

Multiply $\frac{5^{2}}{3^{2}}$ by $1$ .

$\frac{8}{3} + \frac{4}{3} \cdot \frac{5^{2}}{3^{2}} - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.5

Combine.

$\frac{8}{3} + \frac{4 \cdot 5^{2}}{3 \cdot 3^{2}} - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.6

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 7.6.1

Multiply $3$ by $3^{2}$ .

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Step 7.6.1.1

Raise $3$ to the power of $1$ .

$\frac{8}{3} + \frac{4 \cdot 5^{2}}{3^{1} \cdot 3^{2}} - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.6.1.2

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

$\frac{8}{3} + \frac{4 \cdot 5^{2}}{3^{1 + 2}} - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.6.2

Add $1$ and $2$ .

$\frac{8}{3} + \frac{4 \cdot 5^{2}}{3^{3}} - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.7

Raise $5$ to the power of $2$ .

$\frac{8}{3} + \frac{4 \cdot 25}{3^{3}} - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.8

Raise $3$ to the power of $3$ .

$\frac{8}{3} + \frac{4 \cdot 25}{27} - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.9

Multiply $4$ by $25$ .

$\frac{8}{3} + \frac{100}{27} - {(\frac{4}{3})}^{2} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.10

Apply the product rule to $\frac{4}{3}$ .

$\frac{8}{3} + \frac{100}{27} - \frac{4^{2}}{3^{2}} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.11

Raise $4$ to the power of $2$ .

$\frac{8}{3} + \frac{100}{27} - \frac{16}{3^{2}} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.12

Raise $3$ to the power of $2$ .

$\frac{8}{3} + \frac{100}{27} - \frac{16}{9} (- \frac{5}{3}) - 2 (- \frac{5}{3})$

Step 7.13

Multiply $- \frac{16}{9} (- \frac{5}{3})$ .

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Step 7.13.1

Multiply $- 1$ by $- 1$ .

$\frac{8}{3} + \frac{100}{27} + 1 (\frac{16}{9}) \frac{5}{3} - 2 (- \frac{5}{3})$

Step 7.13.2

Multiply $\frac{16}{9}$ by $1$ .

$\frac{8}{3} + \frac{100}{27} + \frac{16}{9} \cdot \frac{5}{3} - 2 (- \frac{5}{3})$

Step 7.13.3

Multiply $\frac{16}{9}$ by $\frac{5}{3}$ .

$\frac{8}{3} + \frac{100}{27} + \frac{16 \cdot 5}{9 \cdot 3} - 2 (- \frac{5}{3})$

Step 7.13.4

Multiply $16$ by $5$ .

$\frac{8}{3} + \frac{100}{27} + \frac{80}{9 \cdot 3} - 2 (- \frac{5}{3})$

Step 7.13.5

Multiply $9$ by $3$ .

$\frac{8}{3} + \frac{100}{27} + \frac{80}{27} - 2 (- \frac{5}{3})$

Step 7.14

Multiply $- 2 (- \frac{5}{3})$ .

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Step 7.14.1

Multiply $- 1$ by $- 2$ .

$\frac{8}{3} + \frac{100}{27} + \frac{80}{27} + 2 (\frac{5}{3})$

Step 7.14.2

Combine $2$ and $\frac{5}{3}$ .

$\frac{8}{3} + \frac{100}{27} + \frac{80}{27} + \frac{2 \cdot 5}{3}$

Step 7.14.3

Multiply $2$ by $5$ .

$\frac{8}{3} + \frac{100}{27} + \frac{80}{27} + \frac{10}{3}$

Step 8

Combine fractions.

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Step 8.1

Combine the numerators over the common denominator.

$\frac{8 + 10}{3} + \frac{100 + 80}{27}$

Step 8.2

Simplify by adding numbers.

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Step 8.2.1

Add $8$ and $10$ .

$\frac{18}{3} + \frac{100 + 80}{27}$

Step 8.2.2

Add $100$ and $80$ .

$\frac{18}{3} + \frac{180}{27}$

Step 9

Simplify each term.

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Step 9.1

Divide $18$ by $3$ .

$6 + \frac{180}{27}$

Step 9.2

Cancel the common factor of $180$ and $27$ .

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Step 9.2.1

Factor $9$ out of $180$ .

$6 + \frac{9 (20)}{27}$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $9$ out of $27$ .

$6 + \frac{9 \cdot 20}{9 \cdot 3}$

Step 9.2.2.2

Cancel the common factor.

$6 + \frac{9 \cdot 20}{9 \cdot 3}$

Step 9.2.2.3

Rewrite the expression.

$6 + \frac{20}{3}$

Step 10

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$6 \cdot \frac{3}{3} + \frac{20}{3}$

Step 11

Combine $6$ and $\frac{3}{3}$ .

$\frac{6 \cdot 3}{3} + \frac{20}{3}$

Step 12

Combine the numerators over the common denominator.

$\frac{6 \cdot 3 + 20}{3}$

Step 13

Simplify the numerator.

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Step 13.1

Multiply $6$ by $3$ .

$\frac{18 + 20}{3}$

Step 13.2

Add $18$ and $20$ .

$\frac{38}{3}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{38}{3}$

Decimal Form:

$12.\overline{6}$

Mixed Number Form:

$12 \frac{2}{3}$