Algebra Examples

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

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(3)}^{4 - k} \cdot {(- \frac{1}{2})}^{k}$

Step 2

Expand the summation.

$\frac{4!}{(4 - 0)! 0!} \cdot {(3)}^{4 - 0} \cdot {(- \frac{1}{2})}^{0} + \frac{4!}{(4 - 1)! 1!} \cdot {(3)}^{4 - 1} \cdot {(- \frac{1}{2})}^{1} + \frac{4!}{(4 - 2)! 2!} \cdot {(3)}^{4 - 2} \cdot {(- \frac{1}{2})}^{2} + \frac{4!}{(4 - 3)! 3!} \cdot {(3)}^{4 - 3} \cdot {(- \frac{1}{2})}^{3} + \frac{4!}{(4 - 4)! 4!} \cdot {(3)}^{4 - 4} \cdot {(- \frac{1}{2})}^{4}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(3)}^{4} \cdot {(- \frac{1}{2})}^{0} + 4 \cdot {(3)}^{3} \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4

Simplify the polynomial result.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply ${(3)}^{4}$ by $1$ .

${(3)}^{4} \cdot {(- \frac{1}{2})}^{0} + 4 \cdot {(3)}^{3} \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.2

Raise $3$ to the power of $4$ .

$81 \cdot {(- \frac{1}{2})}^{0} + 4 \cdot {(3)}^{3} \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.3

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

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Step 4.1.3.1

Apply the product rule to $- \frac{1}{2}$ .

$81 \cdot ({(- 1)}^{0} {(\frac{1}{2})}^{0}) + 4 \cdot {(3)}^{3} \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.3.2

Apply the product rule to $\frac{1}{2}$ .

$81 \cdot ({(- 1)}^{0} \frac{1^{0}}{2^{0}}) + 4 \cdot {(3)}^{3} \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.4

Anything raised to $0$ is $1$ .

$81 \cdot (1 \frac{1^{0}}{2^{0}}) + 4 \cdot {(3)}^{3} \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.5

Multiply $\frac{1^{0}}{2^{0}}$ by $1$ .

$81 \cdot \frac{1^{0}}{2^{0}} + 4 \cdot {(3)}^{3} \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.6

Anything raised to $0$ is $1$ .

$81 \cdot \frac{1}{2^{0}} + 4 \cdot {(3)}^{3} \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.7

Anything raised to $0$ is $1$ .

$81 \cdot \frac{1}{1} + 4 \cdot {(3)}^{3} \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.8

Cancel the common factor of $1$ .

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Step 4.1.8.1

Cancel the common factor.

Step 4.1.8.2

Rewrite the expression.

$81 \cdot 1 + 4 \cdot {(3)}^{3} \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.9

Multiply $81$ by $1$ .

$81 + 4 \cdot {(3)}^{3} \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.10

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

$81 + 4 \cdot 27 \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.11

Multiply $4$ by $27$ .

$81 + 108 \cdot {(- \frac{1}{2})}^{1} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.12

Simplify.

$81 + 108 \cdot (- \frac{1}{2}) + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.13

Cancel the common factor of $2$ .

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Step 4.1.13.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$81 + 108 \cdot \frac{- 1}{2} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.13.2

Factor $2$ out of $108$ .

$81 + 2 (54) \cdot \frac{- 1}{2} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.13.3

Cancel the common factor.

$81 + 2 \cdot 54 \cdot \frac{- 1}{2} + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.13.4

Rewrite the expression.

$81 + 54 \cdot - 1 + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.14

Multiply $54$ by $- 1$ .

$81 - 54 + 6 \cdot {(3)}^{2} \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.15

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

$81 - 54 + 6 \cdot 9 \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.16

Multiply $6$ by $9$ .

$81 - 54 + 54 \cdot {(- \frac{1}{2})}^{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.17

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

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Step 4.1.17.1

Apply the product rule to $- \frac{1}{2}$ .

$81 - 54 + 54 \cdot ({(- 1)}^{2} {(\frac{1}{2})}^{2}) + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.17.2

Apply the product rule to $\frac{1}{2}$ .

$81 - 54 + 54 \cdot ({(- 1)}^{2} \frac{1^{2}}{2^{2}}) + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.18

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

$81 - 54 + 54 \cdot (1 \frac{1^{2}}{2^{2}}) + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.19

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

$81 - 54 + 54 \cdot \frac{1^{2}}{2^{2}} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.20

One to any power is one.

$81 - 54 + 54 \cdot \frac{1}{2^{2}} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.21

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

$81 - 54 + 54 \cdot \frac{1}{4} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.22

Cancel the common factor of $2$ .

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Step 4.1.22.1

Factor $2$ out of $54$ .

$81 - 54 + 2 (27) \cdot \frac{1}{4} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.22.2

Factor $2$ out of $4$ .

$81 - 54 + 2 \cdot 27 \cdot \frac{1}{2 \cdot 2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.22.3

Cancel the common factor.

$81 - 54 + 2 \cdot 27 \cdot \frac{1}{2 \cdot 2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.22.4

Rewrite the expression.

$81 - 54 + 27 \cdot \frac{1}{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.23

Combine $27$ and $\frac{1}{2}$ .

$81 - 54 + \frac{27}{2} + 4 \cdot {(3)}^{1} \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.24

Evaluate the exponent.

$81 - 54 + \frac{27}{2} + 4 \cdot 3 \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.25

Multiply $4$ by $3$ .

$81 - 54 + \frac{27}{2} + 12 \cdot {(- \frac{1}{2})}^{3} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.26

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

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Step 4.1.26.1

Apply the product rule to $- \frac{1}{2}$ .

$81 - 54 + \frac{27}{2} + 12 \cdot ({(- 1)}^{3} {(\frac{1}{2})}^{3}) + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.26.2

Apply the product rule to $\frac{1}{2}$ .

$81 - 54 + \frac{27}{2} + 12 \cdot ({(- 1)}^{3} \frac{1^{3}}{2^{3}}) + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.27

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

$81 - 54 + \frac{27}{2} + 12 \cdot (- \frac{1^{3}}{2^{3}}) + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.28

One to any power is one.

$81 - 54 + \frac{27}{2} + 12 \cdot (- \frac{1}{2^{3}}) + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.29

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

$81 - 54 + \frac{27}{2} + 12 \cdot (- \frac{1}{8}) + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.30

Cancel the common factor of $4$ .

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Step 4.1.30.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$81 - 54 + \frac{27}{2} + 12 \cdot \frac{- 1}{8} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.30.2

Factor $4$ out of $12$ .

$81 - 54 + \frac{27}{2} + 4 (3) \cdot \frac{- 1}{8} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.30.3

Factor $4$ out of $8$ .

$81 - 54 + \frac{27}{2} + 4 \cdot 3 \cdot \frac{- 1}{4 \cdot 2} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.30.4

Cancel the common factor.

$81 - 54 + \frac{27}{2} + 4 \cdot 3 \cdot \frac{- 1}{4 \cdot 2} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.30.5

Rewrite the expression.

$81 - 54 + \frac{27}{2} + 3 \cdot \frac{- 1}{2} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.31

Combine $3$ and $\frac{- 1}{2}$ .

$81 - 54 + \frac{27}{2} + \frac{3 \cdot - 1}{2} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.32

Multiply $3$ by $- 1$ .

$81 - 54 + \frac{27}{2} + \frac{- 3}{2} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.33

Move the negative in front of the fraction.

$81 - 54 + \frac{27}{2} - \frac{3}{2} + 1 \cdot {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.34

Multiply ${(3)}^{0}$ by $1$ .

$81 - 54 + \frac{27}{2} - \frac{3}{2} + {(3)}^{0} \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.35

Anything raised to $0$ is $1$ .

$81 - 54 + \frac{27}{2} - \frac{3}{2} + 1 \cdot {(- \frac{1}{2})}^{4}$

Step 4.1.36

Multiply ${(- \frac{1}{2})}^{4}$ by $1$ .

$81 - 54 + \frac{27}{2} - \frac{3}{2} + {(- \frac{1}{2})}^{4}$

Step 4.1.37

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

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Step 4.1.37.1

Apply the product rule to $- \frac{1}{2}$ .

$81 - 54 + \frac{27}{2} - \frac{3}{2} + {(- 1)}^{4} {(\frac{1}{2})}^{4}$

Step 4.1.37.2

Apply the product rule to $\frac{1}{2}$ .

$81 - 54 + \frac{27}{2} - \frac{3}{2} + {(- 1)}^{4} \frac{1^{4}}{2^{4}}$

Step 4.1.38

Raise $- 1$ to the power of $4$ .

$81 - 54 + \frac{27}{2} - \frac{3}{2} + 1 \frac{1^{4}}{2^{4}}$

Step 4.1.39

Multiply $\frac{1^{4}}{2^{4}}$ by $1$ .

$81 - 54 + \frac{27}{2} - \frac{3}{2} + \frac{1^{4}}{2^{4}}$

Step 4.1.40

One to any power is one.

$81 - 54 + \frac{27}{2} - \frac{3}{2} + \frac{1}{2^{4}}$

Step 4.1.41

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

$81 - 54 + \frac{27}{2} - \frac{3}{2} + \frac{1}{16}$

Step 4.2

Combine fractions.

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Step 4.2.1

Combine the numerators over the common denominator.

$81 - 54 + \frac{27 - 3}{2} + \frac{1}{16}$

Step 4.2.2

Subtract $3$ from $27$ .

$81 - 54 + \frac{24}{2} + \frac{1}{16}$

Step 4.3

Find the common denominator.

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Step 4.3.1

Write $81$ as a fraction with denominator $1$ .

$\frac{81}{1} - 54 + \frac{24}{2} + \frac{1}{16}$

Step 4.3.2

Multiply $\frac{81}{1}$ by $\frac{16}{16}$ .

$\frac{81}{1} \cdot \frac{16}{16} - 54 + \frac{24}{2} + \frac{1}{16}$

Step 4.3.3

Multiply $\frac{81}{1}$ by $\frac{16}{16}$ .

$\frac{81 \cdot 16}{16} - 54 + \frac{24}{2} + \frac{1}{16}$

Step 4.3.4

Write $- 54$ as a fraction with denominator $1$ .

$\frac{81 \cdot 16}{16} + \frac{- 54}{1} + \frac{24}{2} + \frac{1}{16}$

Step 4.3.5

Multiply $\frac{- 54}{1}$ by $\frac{16}{16}$ .

$\frac{81 \cdot 16}{16} + \frac{- 54}{1} \cdot \frac{16}{16} + \frac{24}{2} + \frac{1}{16}$

Step 4.3.6

Multiply $\frac{- 54}{1}$ by $\frac{16}{16}$ .

$\frac{81 \cdot 16}{16} + \frac{- 54 \cdot 16}{16} + \frac{24}{2} + \frac{1}{16}$

Step 4.3.7

Multiply $\frac{24}{2}$ by $\frac{8}{8}$ .

$\frac{81 \cdot 16}{16} + \frac{- 54 \cdot 16}{16} + \frac{24}{2} \cdot \frac{8}{8} + \frac{1}{16}$

Step 4.3.8

Multiply $\frac{24}{2}$ by $\frac{8}{8}$ .

$\frac{81 \cdot 16}{16} + \frac{- 54 \cdot 16}{16} + \frac{24 \cdot 8}{2 \cdot 8} + \frac{1}{16}$

Step 4.3.9

Multiply $2$ by $8$ .

$\frac{81 \cdot 16}{16} + \frac{- 54 \cdot 16}{16} + \frac{24 \cdot 8}{16} + \frac{1}{16}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{81 \cdot 16 - 54 \cdot 16 + 24 \cdot 8 + 1}{16}$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $81$ by $16$ .

$\frac{1296 - 54 \cdot 16 + 24 \cdot 8 + 1}{16}$

Step 4.5.2

Multiply $- 54$ by $16$ .

$\frac{1296 - 864 + 24 \cdot 8 + 1}{16}$

Step 4.5.3

Multiply $24$ by $8$ .

$\frac{1296 - 864 + 192 + 1}{16}$

Step 4.6

Simplify by adding and subtracting.

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Step 4.6.1

Subtract $864$ from $1296$ .

$\frac{432 + 192 + 1}{16}$

Step 4.6.2

Add $432$ and $192$ .

$\frac{624 + 1}{16}$

Step 4.6.3

Add $624$ and $1$ .

$\frac{625}{16}$