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Popular >

simplify sqrt(32)^5

Solution

simplify

32^{5}

Solution

40962

+1

Decimal Notation

5792.61875 \dots

Solution steps

(32)^{5}

Apply exponent rule:

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

(32)^{4 + 1} = (32)^{4} 32

= (32)^{4} 32

(32)^{4} = ((32)^{2})^{2}

= ((32)^{2})^{2} 32

Apply exponent rule:

(32)^{2} = 32

= 3 2^{2} 32

Apply radical rule:

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

32 = 3 2^{\frac{1}{2}}

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

Apply exponent rule:

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

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

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

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

= 3 2^{\frac{5}{2}}

Prime factorization of

32 : 2^{5}

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

Apply exponent rule:

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

2^{5} = 2^{4} \cdot 2

= (2^{4} \cdot 2)^{\frac{5}{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(2^{4} \cdot 2)^{\frac{5}{2}} = (2^{4})^{\frac{5}{2}} \cdot 2^{\frac{5}{2}}

= (2^{4})^{\frac{5}{2}} \cdot 2^{\frac{5}{2}}

(2^{4})^{\frac{5}{2}} = 1024

= 1024 \cdot 2^{\frac{5}{2}}

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

= 2^{2} \cdot 10242

Factor integer

1024 = 2^{10}

= 2^{10} \cdot 2^{2} 2

Apply exponent rule:

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

2^{10} \cdot 2^{2} = 2^{10 + 2}

= 2^{10 + 2} 2

Add the numbers:

10 + 2 = 12

= 2^{12} 2

2^{12} = 4096

= 40962

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Frequently Asked Questions (FAQ)

What is sqrt(32)^5 ?
The solution to sqrt(32)^5 is 4096sqrt(2)