Evaluate (1-1/(20^2))(1-1/(19^2))(1-1/(18^2))

$(1 - \frac{1}{20^{2}}) (1 - \frac{1}{19^{2}}) (1 - \frac{1}{18^{2}})$

Step 1

Simplify the expression.

Raise $20$ to the power of $2$ .

$(1 - \frac{1}{400}) (1 - \frac{1}{19^{2}}) (1 - \frac{1}{18^{2}})$

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

$(\frac{400}{400} - \frac{1}{400}) (1 - \frac{1}{19^{2}}) (1 - \frac{1}{18^{2}})$

Combine the numerators over the common denominator.

$\frac{400 - 1}{400} (1 - \frac{1}{19^{2}}) (1 - \frac{1}{18^{2}})$

Subtract $1$ from $400$ .

$\frac{399}{400} (1 - \frac{1}{19^{2}}) (1 - \frac{1}{18^{2}})$

Raise $19$ to the power of $2$ .

$\frac{399}{400} (1 - \frac{1}{361}) (1 - \frac{1}{18^{2}})$

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

$\frac{399}{400} (\frac{361}{361} - \frac{1}{361}) (1 - \frac{1}{18^{2}})$

Combine the numerators over the common denominator.

$\frac{399}{400} \cdot \frac{361 - 1}{361} (1 - \frac{1}{18^{2}})$

Subtract $1$ from $361$ .

$\frac{399}{400} \cdot \frac{360}{361} (1 - \frac{1}{18^{2}})$

Step 2

Cancel the common factor of $19$ .

Factor $19$ out of $399$ .

$\frac{19 (21)}{400} \cdot \frac{360}{361} (1 - \frac{1}{18^{2}})$

Factor $19$ out of $361$ .

$\frac{19 \cdot 21}{400} \cdot \frac{360}{19 \cdot 19} (1 - \frac{1}{18^{2}})$

Cancel the common factor.

$\frac{19 \cdot 21}{400} \cdot \frac{360}{19 \cdot 19} (1 - \frac{1}{18^{2}})$

Rewrite the expression.

$\frac{21}{400} \cdot \frac{360}{19} (1 - \frac{1}{18^{2}})$

Step 3

Cancel the common factor of $40$ .

Factor $40$ out of $400$ .

$\frac{21}{40 (10)} \cdot \frac{360}{19} (1 - \frac{1}{18^{2}})$

Factor $40$ out of $360$ .

$\frac{21}{40 \cdot 10} \cdot \frac{40 \cdot 9}{19} (1 - \frac{1}{18^{2}})$

Cancel the common factor.

$\frac{21}{40 \cdot 10} \cdot \frac{40 \cdot 9}{19} (1 - \frac{1}{18^{2}})$

Rewrite the expression.

$\frac{21}{10} \cdot \frac{9}{19} (1 - \frac{1}{18^{2}})$

Step 4

Multiply $\frac{21}{10}$ by $\frac{9}{19}$ .

$\frac{21 \cdot 9}{10 \cdot 19} (1 - \frac{1}{18^{2}})$

Step 5

Simplify the expression.

Multiply $21$ by $9$ .

$\frac{189}{10 \cdot 19} (1 - \frac{1}{18^{2}})$

Multiply $10$ by $19$ .

$\frac{189}{190} (1 - \frac{1}{18^{2}})$

Raise $18$ to the power of $2$ .

$\frac{189}{190} (1 - \frac{1}{324})$

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

$\frac{189}{190} (\frac{324}{324} - \frac{1}{324})$

Combine the numerators over the common denominator.

$\frac{189}{190} \cdot \frac{324 - 1}{324}$

Subtract $1$ from $324$ .

$\frac{189}{190} \cdot \frac{323}{324}$

Step 6

Cancel the common factor of $27$ .

Factor $27$ out of $189$ .

$\frac{27 (7)}{190} \cdot \frac{323}{324}$

Factor $27$ out of $324$ .

$\frac{27 \cdot 7}{190} \cdot \frac{323}{27 \cdot 12}$

Cancel the common factor.

$\frac{27 \cdot 7}{190} \cdot \frac{323}{27 \cdot 12}$

Rewrite the expression.

$\frac{7}{190} \cdot \frac{323}{12}$

Step 7

Cancel the common factor of $19$ .

Factor $19$ out of $190$ .

$\frac{7}{19 (10)} \cdot \frac{323}{12}$

Factor $19$ out of $323$ .

$\frac{7}{19 \cdot 10} \cdot \frac{19 \cdot 17}{12}$

Cancel the common factor.

$\frac{7}{19 \cdot 10} \cdot \frac{19 \cdot 17}{12}$

Rewrite the expression.

$\frac{7}{10} \cdot \frac{17}{12}$

Step 8

Multiply $\frac{7}{10}$ by $\frac{17}{12}$ .

$\frac{7 \cdot 17}{10 \cdot 12}$

Step 9

Multiply.

Multiply $7$ by $17$ .

$\frac{119}{10 \cdot 12}$

Multiply $10$ by $12$ .

$\frac{119}{120}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{119}{120}$

Decimal Form:

$0.991\overline{6}$

Evaluate (1-1/(20^2))(1-1/(19^2))(1-1/(18^2))

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