What is the value of 8/15*0.58*tan(45)*sqrt(2*9.81)*0.1^{5/2} ?

The value of 8/15*0.58*tan(45)*sqrt(2*9.81)*0.1^{5/2} is (29*109^{1/2})/(31250*5^{1/2)}

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8/150.58tan(45)sqrt(29.81)*0.1^{5/2}

Solution

\frac{8}{15} \cdot 0.58 \cdot tan (4 5^{\circ}) \cdot 2 \cdot 9.81 \cdot 0. 1^{\frac{5}{2}}

Solution

\frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

Decimal Notation

0.00433 \dots

Solution steps

\frac{8}{15} \cdot 0.58 tan (4 5^{\circ}) 2 \cdot 9.81 \cdot 0. 1^{\frac{5}{2}}

= \frac{8}{15} \cdot \frac{29}{50} tan (4 5^{\circ}) 2 \cdot \frac{981}{100} (\frac{1}{10})^{\frac{5}{2}}

Use the following trivial identity:

tan (4 5^{\circ}) = 1

tan (4 5^{\circ})

tan (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 1

= \frac{8}{15} \cdot \frac{29}{50} \cdot 1 \cdot 2 \cdot \frac{981}{100} (\frac{1}{10})^{\frac{5}{2}}

Simplify

\frac{8}{15} \cdot \frac{29}{50} \cdot 1 \cdot 2 \cdot \frac{981}{100} (\frac{1}{10})^{\frac{5}{2}} : \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

\frac{8}{15} \cdot \frac{29}{50} \cdot 1 \cdot 2 \cdot \frac{981}{100} (\frac{1}{10})^{\frac{5}{2}}

2 \cdot \frac{981}{100} = \frac{3 \cdot 10 9 ^{\frac{1}{2}}}{5 \cdot 2 ^{\frac{1}{2}}}

2 \cdot \frac{981}{100}

Multiply

2 \cdot \frac{981}{100} : \frac{981}{50}

2 \cdot \frac{981}{100}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{981 \cdot 2}{100}

Multiply the numbers:

981 \cdot 2 = 1962

= \frac{1962}{100}

Cancel the common factor:

2

= \frac{981}{50}

= \frac{981}{50}

Apply radical rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}},

assuming

a \geq 0, b \geq 0

= \frac{98 1 ^{\frac{1}{2}}}{5 0 ^{\frac{1}{2}}}

5 0^{\frac{1}{2}} = 5 \cdot 2^{\frac{1}{2}}

5 0^{\frac{1}{2}}

Prime factorization of

50 : 2 \cdot 5^{2}

50

50

divides by

250 = 25 \cdot 2

= 2 \cdot 25

25

divides by

525 = 5 \cdot 5

= 2 \cdot 5 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 5 \cdot 5

= 2 \cdot 5^{2}

= (5^{2} \cdot 2)^{\frac{1}{2}}

Apply exponent rule:

a \cdot b^{c} = a^{c} b^{c}

= 2^{\frac{1}{2}} (5^{2})^{\frac{1}{2}}

Apply exponent rule:

(5^{2})^{\frac{1}{2}} = 5^{\frac{2}{2}} = 5

= 5 \cdot 2^{\frac{1}{2}}

= \frac{98 1 ^{\frac{1}{2}}}{5 \cdot 2 ^{\frac{1}{2}}}

98 1^{\frac{1}{2}} = 3 \cdot 10 9^{\frac{1}{2}}

98 1^{\frac{1}{2}}

Prime factorization of

981 : 3^{2} \cdot 109

981

981

divides by

3981 = 327 \cdot 3

= 3 \cdot 327

327

divides by

3327 = 109 \cdot 3

= 3 \cdot 3 \cdot 109

3, 109

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 3 \cdot 109

= 3^{2} \cdot 109

= (3^{2} \cdot 109)^{\frac{1}{2}}

Apply exponent rule:

a \cdot b^{c} = a^{c} b^{c}

= 10 9^{\frac{1}{2}} (3^{2})^{\frac{1}{2}}

Apply exponent rule:

(3^{2})^{\frac{1}{2}} = 3^{\frac{2}{2}} = 3

= 3 \cdot 10 9^{\frac{1}{2}}

= \frac{3 \cdot 10 9 ^{\frac{1}{2}}}{5 \cdot 2 ^{\frac{1}{2}}}

= 1 \cdot \frac{8}{15} \cdot \frac{29}{50} \cdot \frac{3 \cdot 10 9 ^{\frac{1}{2}}}{5 \cdot 2 ^{\frac{1}{2}}} (\frac{1}{10})^{\frac{5}{2}}

Multiply fractions:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= 1 \cdot \frac{8 \cdot 29 \cdot 3 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{15 \cdot 50 \cdot 5 \cdot 2 ^{\frac{1}{2}}}

\frac{8 \cdot 29 \cdot 3 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{15 \cdot 50 \cdot 5 \cdot 2 ^{\frac{1}{2}}} = \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

\frac{8 \cdot 29 \cdot 3 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{15 \cdot 50 \cdot 5 \cdot 2 ^{\frac{1}{2}}}

Multiply the numbers:

8 \cdot 29 \cdot 3 = 696

= \frac{696 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{15 \cdot 50 \cdot 5 \cdot 2 ^{\frac{1}{2}}}

Multiply the numbers:

15 \cdot 50 \cdot 5 = 3750

= \frac{696 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{3750 \cdot 2 ^{\frac{1}{2}}}

Cancel the common factor:

6

= \frac{116 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625 \cdot 2 ^{\frac{1}{2}}}

Factor

116 : 2^{2} \cdot 29

Factor

116 = 2^{2} \cdot 29

= \frac{2 ^{2} \cdot 29 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625 \cdot 2 ^{\frac{1}{2}}}

Cancel

\frac{2 ^{2} \cdot 29 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625 \cdot 2 ^{\frac{1}{2}}} : \frac{29 \cdot 2 ^{\frac{3}{2}} \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625}

\frac{2 ^{2} \cdot 29 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625 \cdot 2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{2}}{2 ^{\frac{1}{2}}} = 2^{2 - \frac{1}{2}}

= \frac{29 \cdot 10 9 ^{\frac{1}{2}} \cdot 2 ^{- \frac{1}{2} + 2} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{29 \cdot 2 ^{\frac{3}{2}} \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625}

= \frac{29 \cdot 2 ^{\frac{3}{2}} \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625}

2^{\frac{3}{2}} = 2 \cdot 2^{\frac{1}{2}}

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 2 \cdot 2^{\frac{1}{2}}

= \frac{29 \cdot 2 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625}

(\frac{1}{10})^{\frac{5}{2}} = \frac{1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

(\frac{1}{10})^{\frac{5}{2}}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{\frac{5}{2}}}{1 0 ^{\frac{5}{2}}}

1 0^{\frac{5}{2}} = 1 0^{2} \cdot 1 0^{\frac{1}{2}}

1 0^{\frac{5}{2}}

1 0^{\frac{5}{2}} = 1 0^{2 + \frac{1}{2}}

= 1 0^{2 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 1 0^{2} \cdot 1 0^{\frac{1}{2}}

= 1 0^{2} \cdot 1 0^{\frac{1}{2}}

1^{\frac{5}{2}} = 1^{2} \cdot 1^{\frac{1}{2}}

1^{\frac{5}{2}}

1^{\frac{5}{2}} = 1^{2 + \frac{1}{2}}

= 1^{2 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 1^{2} \cdot 1^{\frac{1}{2}}

= \frac{1 ^{2} \cdot 1 ^{\frac{1}{2}}}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

Apply rule

1^{a} = 1

1^{2} = 1, 1^{\frac{1}{2}} = 1

= \frac{1 \cdot 1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

= \frac{29 \cdot 2 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}} \cdot \frac{1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}}{625}

Multiply the numbers:

29 \cdot 2 = 58

= \frac{58 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}} \cdot \frac{1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}}{625}

Multiply

58 \cdot 2^{\frac{1}{2}} \cdot 10 9^{\frac{1}{2}} \cdot \frac{1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}} : \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{50 \cdot 5 ^{\frac{1}{2}}}

58 \cdot 2^{\frac{1}{2}} \cdot 10 9^{\frac{1}{2}} \cdot \frac{1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 58 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}}}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

Multiply the numbers:

1 \cdot 58 = 58

= \frac{58 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}}}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

Factor

1 0^{2} : 2^{2} \cdot 5^{2}

Factor

10 = 2 \cdot 5

= (2 \cdot 5)^{2}

Apply exponent rule:

(ab)^{c} = a^{c} b^{c}

= 2^{2} \cdot 5^{2}

Factor

1 0^{\frac{1}{2}} : 2^{\frac{1}{2}} \cdot 5^{\frac{1}{2}}

Factor

10 = 2 \cdot 5

= (2 \cdot 5)^{\frac{1}{2}}

Apply exponent rule:

(ab)^{c} = a^{c} b^{c}

= 2^{\frac{1}{2}} \cdot 5^{\frac{1}{2}}

= \frac{58 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}}}{2 ^{2} \cdot 5 ^{2} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}}

Cancel

\frac{58 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}}}{2 ^{2} \cdot 5 ^{2} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}} : \frac{58 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 2 ^{\frac{3}{2}} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}}

\frac{58 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}}}{2 ^{2} \cdot 5 ^{2} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{58 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}} \cdot 2 ^{- \frac{1}{2} + 2}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{58 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 2 ^{\frac{3}{2}} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}}

= \frac{58 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 2 ^{\frac{3}{2}} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}}

5^{2} \cdot 2^{\frac{3}{2}} \cdot 2^{\frac{1}{2}} \cdot 5^{\frac{1}{2}} = 5^{2} \cdot 5^{\frac{1}{2}} \cdot 2^{\frac{3}{2} + \frac{1}{2}}

5^{2} \cdot 2^{\frac{3}{2}} \cdot 2^{\frac{1}{2}} \cdot 5^{\frac{1}{2}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{3}{2}} \cdot 2^{\frac{1}{2}} = 2^{\frac{3}{2} + \frac{1}{2}}

= 5^{2} \cdot 2^{\frac{3}{2} + \frac{1}{2}} \cdot 5^{\frac{1}{2}}

= \frac{58 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 5 ^{\frac{1}{2}} \cdot 2 ^{\frac{3}{2} + \frac{1}{2}}}

Factor

58 : 2 \cdot 29

Factor

58 = 2 \cdot 29

= \frac{2 \cdot 29 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 5 ^{\frac{1}{2}} \cdot 2 ^{\frac{3}{2} + \frac{1}{2}}}

Cancel the common factor:

2

= \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 2 \cdot 5 ^{\frac{1}{2}}}

5^{2} \cdot 2 \cdot 5^{\frac{1}{2}} = 50 \cdot 5^{\frac{1}{2}}

5^{2} \cdot 2 \cdot 5^{\frac{1}{2}}

5^{2} = 25

= 25 \cdot 2 \cdot 5^{\frac{1}{2}}

Multiply the numbers:

25 \cdot 2 = 50

= 50 \cdot 5^{\frac{1}{2}}

= \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{50 \cdot 5 ^{\frac{1}{2}}}

= \frac{\frac{29 \cdot 10 9 ^{\frac{1}{2}}}{50 \cdot 5 ^{\frac{1}{2}}}}{625}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{50 \cdot 5 ^{\frac{1}{2}} \cdot 625}

Multiply the numbers:

50 \cdot 625 = 31250

= \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

= 1 \cdot \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

Multiply:

1 \cdot \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}} = \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

= \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

= \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

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Frequently Asked Questions (FAQ)

What is the value of 8/15*0.58*tan(45)*sqrt(2*9.81)*0.1^{5/2} ?
The value of 8/15*0.58*tan(45)*sqrt(2*9.81)*0.1^{5/2} is (29*109^{1/2})/(31250*5^{1/2)}