What is the value of arccos(4/(sqrt(5*\sqrt{5))}) ?

The value of arccos(4/(sqrt(5*\sqrt{5))}) is No Solution for x\in\mathbb{R}

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Popular Trigonometry >

arccos(4/(sqrt(5*\sqrt{5))})

Solution

arccos (\frac{4}{5 \cdot 5})

Solution

No Solution for x \in R

Solution steps

arccos (\frac{4}{5 5})

Simplify:

\frac{4}{5 5}

55 = 5^{\frac{3}{4}}

55

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 55

Apply radical rule:

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

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

Apply exponent rule:

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

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

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

\frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= 5^{\frac{1}{4}}

Apply exponent rule:

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

= 5^{\frac{1}{2} + \frac{1}{4}}

Join

\frac{1}{2} + \frac{1}{4} : \frac{3}{4}

\frac{1}{2} + \frac{1}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

4

For

\frac{1}{2} :

multiply the denominator and numerator by

2

\frac{1}{2} = \frac{1 \cdot 2}{2 \cdot 2} = \frac{2}{4}

= \frac{2}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{2 + 1}{4}

Add the numbers:

2 + 1 = 3

= \frac{3}{4}

= 5^{\frac{3}{4}}

= \frac{4}{5 ^{\frac{3}{4}}}

Rationalize

\frac{4}{5 ^{\frac{3}{4}}}

Multiply by the conjugate

Apply exponent rule:

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

= 5^{\frac{3}{4} + \frac{1}{4}}

Join

\frac{3}{4} + \frac{1}{4} : 1

\frac{3}{4} + \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 1}{4}

Add the numbers:

3 + 1 = 4

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

= 5^{1}

Apply rule

a^{1} = a

= 5

Real domain for

arccos (x)

- 1 \leq x \leq 1

No Solution for x \in R

Popular Examples

cos(295)

2sin^2(pi/4)

arccos(cos(-(3pi)/2))

3e^0cos(pi/3)

csc((-3pi)/4)

Frequently Asked Questions (FAQ)

What is the value of arccos(4/(sqrt(5*\sqrt{5))}) ?
The value of arccos(4/(sqrt(5*\sqrt{5))}) is No Solution for x\in\mathbb{R}