Pre-Algebra Examples

$| ({(\frac{6}{3})}^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{5^{2}}{2})) + 10^{2} \cdot {(\frac{4}{5})}^{2} \div \frac{2^{6}}{3} | \div (\frac{36}{15} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2})$

Step 1

Rewrite the division as a fraction.

$\frac{| ({(\frac{6}{3})}^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{5^{2}}{2})) + 10^{2} \cdot {(\frac{4}{5})}^{2} \div \frac{2^{6}}{3} |}{\frac{36}{15} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $6$ and $3$ .

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Step 2.1.1

Factor $3$ out of $6$ .

$\frac{| {(\frac{3 \cdot 2}{3})}^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{5^{2}}{2}) + 10^{2} \cdot {(\frac{4}{5})}^{2} \div \frac{2^{6}}{3} |}{\frac{36}{15} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $3$ out of $3$ .

$\frac{| {(\frac{3 \cdot 2}{3 (1)})}^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{5^{2}}{2}) + 10^{2} \cdot {(\frac{4}{5})}^{2} \div \frac{2^{6}}{3} |}{\frac{36}{15} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 2.1.2.2

Cancel the common factor.

$\frac{| {(\frac{3 \cdot 2}{3 \cdot 1})}^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{5^{2}}{2}) + 10^{2} \cdot {(\frac{4}{5})}^{2} \div \frac{2^{6}}{3} |}{\frac{36}{15} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 2.1.2.3

Rewrite the expression.

$\frac{| {(\frac{2}{1})}^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{5^{2}}{2}) + 10^{2} \cdot {(\frac{4}{5})}^{2} \div \frac{2^{6}}{3} |}{\frac{36}{15} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 2.1.2.4

Divide $2$ by $1$ .

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{5^{2}}{2}) + 10^{2} \cdot {(\frac{4}{5})}^{2} \div \frac{2^{6}}{3} |}{\frac{36}{15} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 2.2

Cancel the common factor of $36$ and $15$ .

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Step 2.2.1

Factor $3$ out of $36$ .

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{5^{2}}{2}) + 10^{2} \cdot {(\frac{4}{5})}^{2} \div \frac{2^{6}}{3} |}{\frac{3 (12)}{15} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $3$ out of $15$ .

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{5^{2}}{2}) + 10^{2} \cdot {(\frac{4}{5})}^{2} \div \frac{2^{6}}{3} |}{\frac{3 \cdot 12}{3 \cdot 5} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 2.2.2.2

Cancel the common factor.

Step 2.2.2.3

Rewrite the expression.

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{5^{2}}{2}) + 10^{2} \cdot {(\frac{4}{5})}^{2} \div \frac{2^{6}}{3} |}{\frac{12}{5} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 2.3

Evaluate the exponents.

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Step 2.3.1

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

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{25}{2}) + 10^{2} \cdot {(\frac{4}{5})}^{2} \div \frac{2^{6}}{3} |}{\frac{12}{5} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 2.3.2

Raise $2$ to the power of $6$ .

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{25}{2}) + 10^{2} \cdot {(\frac{4}{5})}^{2} \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 3

Simplify the numerator.

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Step 3.1

Apply the product rule to $\frac{4}{5}$ .

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{25}{2}) + 10^{2} \frac{4^{2}}{5^{2}} \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 3.2

Raise $10$ to the power of $2$ .

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{25}{2}) + 100 \frac{4^{2}}{5^{2}} \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 3.3

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

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{25}{2}) + 100 \frac{16}{5^{2}} \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 3.4

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

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{25}{2}) + 100 (\frac{16}{25}) \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 4

Combine fractions.

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Step 4.1

Combine $100$ and $\frac{16}{25}$ .

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{25}{2}) + \frac{100 \cdot 16}{25} \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 4.2

Simplify the expression.

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Step 4.2.1

Multiply $100$ by $16$ .

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{25}{2}) + \frac{1600}{25} \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 4.2.2

Divide $1600$ by $25$ .

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{25}{2}) + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{2^{2}})}^{2} \cdot \frac{1}{2}}$

Step 4.2.3

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

$\frac{| 2^{2} - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{25}{2}) + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{| 4 - (\frac{1}{2} + \frac{1}{16} \div \frac{1}{2} \cdot \frac{25}{2}) + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.2

Simplify each term.

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Step 5.2.1

To divide by a fraction, multiply by its reciprocal.

$\frac{| 4 - (\frac{1}{2} + \frac{1}{16} \cdot 2 \cdot \frac{25}{2}) + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.2.2

Cancel the common factor of $2$ .

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Step 5.2.2.1

Factor $2$ out of $\frac{1}{16} \cdot 2$ .

$\frac{| 4 - (\frac{1}{2} + 2 \cdot \frac{1}{16} \cdot \frac{25}{2}) + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.2.2.2

Cancel the common factor.

$\frac{| 4 - (\frac{1}{2} + 2 \cdot \frac{1}{16} \cdot \frac{25}{2}) + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.2.2.3

Rewrite the expression.

$\frac{| 4 - (\frac{1}{2} + \frac{1}{16} \cdot 25) + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.2.3

Combine $\frac{1}{16}$ and $25$ .

$\frac{| 4 - (\frac{1}{2} + \frac{25}{16}) + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{| 4 - (\frac{1}{2} \cdot \frac{8}{8} + \frac{25}{16}) + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.4

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.1

Multiply $\frac{1}{2}$ by $\frac{8}{8}$ .

$\frac{| 4 - (\frac{8}{2 \cdot 8} + \frac{25}{16}) + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.4.2

Multiply $2$ by $8$ .

$\frac{| 4 - (\frac{8}{16} + \frac{25}{16}) + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.5

Combine the numerators over the common denominator.

$\frac{| 4 - \frac{8 + 25}{16} + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.6

Add $8$ and $25$ .

$\frac{| 4 - \frac{33}{16} + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.7

To write $4$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{| 4 \cdot \frac{16}{16} - \frac{33}{16} + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.8

Combine $4$ and $\frac{16}{16}$ .

$\frac{| \frac{4 \cdot 16}{16} - \frac{33}{16} + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.9

Combine the numerators over the common denominator.

$\frac{| \frac{4 \cdot 16 - 33}{16} + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.10

Simplify the numerator.

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Step 5.10.1

Multiply $4$ by $16$ .

$\frac{| \frac{64 - 33}{16} + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.10.2

Subtract $33$ from $64$ .

$\frac{| \frac{31}{16} + 64 \div \frac{64}{3} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.11

Simplify each term.

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Step 5.11.1

To divide by a fraction, multiply by its reciprocal.

$\frac{| \frac{31}{16} + 64 (\frac{3}{64}) |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.11.2

Cancel the common factor of $64$ .

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Step 5.11.2.1

Cancel the common factor.

$\frac{| \frac{31}{16} + 64 (\frac{3}{64}) |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.11.2.2

Rewrite the expression.

$\frac{| \frac{31}{16} + 3 |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.12

To write $3$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{| \frac{31}{16} + 3 \cdot \frac{16}{16} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.13

Combine $3$ and $\frac{16}{16}$ .

$\frac{| \frac{31}{16} + \frac{3 \cdot 16}{16} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.14

Combine the numerators over the common denominator.

$\frac{| \frac{31 + 3 \cdot 16}{16} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.15

Simplify the numerator.

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Step 5.15.1

Multiply $3$ by $16$ .

$\frac{| \frac{31 + 48}{16} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.15.2

Add $31$ and $48$ .

$\frac{| \frac{79}{16} |}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 5.16

$\frac{79}{16}$ is approximately $4.9375$ which is positive so remove the absolute value

$\frac{\frac{79}{16}}{\frac{12}{5} + {(\frac{3}{4})}^{2} \cdot \frac{1}{2}}$

Step 6

Simplify the denominator.

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Step 6.1

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

$\frac{\frac{79}{16}}{\frac{12}{5} + \frac{3^{2}}{4^{2}} \cdot \frac{1}{2}}$

Step 6.2

Combine.

$\frac{\frac{79}{16}}{\frac{12}{5} + \frac{3^{2} \cdot 1}{4^{2} \cdot 2}}$

Step 6.3

Multiply $3^{2}$ by $1$ .

$\frac{\frac{79}{16}}{\frac{12}{5} + \frac{3^{2}}{4^{2} \cdot 2}}$

Step 6.4

Simplify the denominator.

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Step 6.4.1

Rewrite $4$ as $2^{2}$ .

$\frac{\frac{79}{16}}{\frac{12}{5} + \frac{3^{2}}{{(2^{2})}^{2} \cdot 2}}$

Step 6.4.2

Multiply the exponents in ${(2^{2})}^{2}$ .

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Step 6.4.2.1

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

$\frac{\frac{79}{16}}{\frac{12}{5} + \frac{3^{2}}{2^{2 \cdot 2} \cdot 2}}$

Step 6.4.2.2

Multiply $2$ by $2$ .

$\frac{\frac{79}{16}}{\frac{12}{5} + \frac{3^{2}}{2^{4} \cdot 2}}$

Step 6.4.3

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

$\frac{\frac{79}{16}}{\frac{12}{5} + \frac{3^{2}}{2^{4 + 1}}}$

Step 6.4.4

Add $4$ and $1$ .

$\frac{\frac{79}{16}}{\frac{12}{5} + \frac{3^{2}}{2^{5}}}$

Step 6.5

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

$\frac{\frac{79}{16}}{\frac{12}{5} + \frac{9}{2^{5}}}$

Step 6.6

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

$\frac{\frac{79}{16}}{\frac{12}{5} + \frac{9}{32}}$

Step 6.7

To write $\frac{12}{5}$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$\frac{\frac{79}{16}}{\frac{12}{5} \cdot \frac{32}{32} + \frac{9}{32}}$

Step 6.8

To write $\frac{9}{32}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{\frac{79}{16}}{\frac{12}{5} \cdot \frac{32}{32} + \frac{9}{32} \cdot \frac{5}{5}}$

Step 6.9

Write each expression with a common denominator of $160$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.9.1

Multiply $\frac{12}{5}$ by $\frac{32}{32}$ .

$\frac{\frac{79}{16}}{\frac{12 \cdot 32}{5 \cdot 32} + \frac{9}{32} \cdot \frac{5}{5}}$

Step 6.9.2

Multiply $5$ by $32$ .

$\frac{\frac{79}{16}}{\frac{12 \cdot 32}{160} + \frac{9}{32} \cdot \frac{5}{5}}$

Step 6.9.3

Multiply $\frac{9}{32}$ by $\frac{5}{5}$ .

$\frac{\frac{79}{16}}{\frac{12 \cdot 32}{160} + \frac{9 \cdot 5}{32 \cdot 5}}$

Step 6.9.4

Multiply $32$ by $5$ .

$\frac{\frac{79}{16}}{\frac{12 \cdot 32}{160} + \frac{9 \cdot 5}{160}}$

Step 6.10

Combine the numerators over the common denominator.

$\frac{\frac{79}{16}}{\frac{12 \cdot 32 + 9 \cdot 5}{160}}$

Step 6.11

Simplify the numerator.

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Step 6.11.1

Multiply $12$ by $32$ .

$\frac{\frac{79}{16}}{\frac{384 + 9 \cdot 5}{160}}$

Step 6.11.2

Multiply $9$ by $5$ .

$\frac{\frac{79}{16}}{\frac{384 + 45}{160}}$

Step 6.11.3

Add $384$ and $45$ .

$\frac{\frac{79}{16}}{\frac{429}{160}}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\frac{79}{16} \cdot \frac{160}{429}$

Step 8

Cancel the common factor of $16$ .

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Step 8.1

Factor $16$ out of $160$ .

$\frac{79}{16} \cdot \frac{16 (10)}{429}$

Step 8.2

Cancel the common factor.

$\frac{79}{16} \cdot \frac{16 \cdot 10}{429}$

Step 8.3

Rewrite the expression.

$79 (\frac{10}{429})$

Step 9

Combine $79$ and $\frac{10}{429}$ .

$\frac{79 \cdot 10}{429}$

Step 10

Multiply $79$ by $10$ .

$\frac{790}{429}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{790}{429}$

Decimal Form:

$1.\overline{841491}$

Mixed Number Form:

$1 \frac{361}{429}$