Simplify ((16)/(81))^(1)/(4) / ((9)/(16))^-(1)/(2)

Mathematics
Simplify ((16)/(81))^(1)/(4) / ((9)/(16))^-(1)/(2)

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Answer

12\frac{1}{2}

Step 1: Simplify (1681)14\left( \frac{16}{81} \right)^{\frac{1}{4}}.

Note that 16=2416 = 2^{4} and 81=3481 = 3^{4}.

1681=2434=(23)4\frac{16}{81} = \frac{2^{4}}{3^{4}} = \left( \frac{2}{3} \right)^{4} (1681)14=[(23)4]14=(23)4⋅14=(23)1=23\left( \frac{16}{81} \right)^{\frac{1}{4}} = \left[ \left( \frac{2}{3} \right)^{4} \right]^{\frac{1}{4}} = \left( \frac{2}{3} \right)^{4 \cdot \frac{1}{4}} = \left( \frac{2}{3} \right)^{1} = \frac{2}{3}

Step 2: Simplify (916)12\left( \frac{9}{16} \right)^{\frac{1}{2}}.

Note that 9=329 = 3^{2} and 16=4216 = 4^{2}.

916=3242=(34)2\frac{9}{16} = \frac{3^{2}}{4^{2}} = \left( \frac{3}{4} \right)^{2} (916)12=[(34)2]12=(34)2⋅12=(34)1=34\left( \frac{9}{16} \right)^{\frac{1}{2}} = \left[ \left( \frac{3}{4} \right)^{2} \right]^{\frac{1}{2}} = \left( \frac{3}{4} \right)^{2 \cdot \frac{1}{2}} = \left( \frac{3}{4} \right)^{1} = \frac{3}{4}

Step 3: Multiply the results.

23⋅34\frac{2}{3} \cdot \frac{3}{4}

Substitute numerators and denominators:

2⋅33⋅4\frac{2 \cdot 3}{3 \cdot 4}

Cancel common factor 33:

24\frac{2}{4}

Simplify by dividing numerator and denominator by 22:

2÷24÷2=12\frac{2 \div 2}{4 \div 2} = \frac{1}{2}

\frac{1{2}}

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Quick Answer

Simplify ( (16)/(81) )^(1)/(4). Note that 16 = 2^4 and 81 = 3^4.

Simplify ((16)/(81))^(1)/(4) / ((9)/(16))^-(1)/(2)
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
Step 1: Simplify ( (16)/(81) )^(1)/(4). Note that 16 = 2^4 and 81 = 3^4. (16)/(81) = 2^43^4 = ( (2)/(3) )^4 ( (16)/(81) )^(1)/(4) = [ ( (2)/(3) )^4 ]^(1)/(4) = ( (2)/(3) )^4 · (1)/(4) = ( (2)/(3) )^1 = (2)/(3) Step 2: Simplify ( (9)/(16) )^(1)/(2). Note that 9 = 3^2 and 16 = 4^2. (9)/(16) = 3^24^2 = ( (3)/(4) )^2 ( (9)/(16) )^(1)/(2) = [ ( (3)/(4) )^2 ]^(1)/(2) = ( (3)/(4) )^2 · (1)/(2) = ( (3)/(4) )^1 = (3)/(4) Step 3: Multiply the results. (2)/(3) · (3)/(4) Substitute numerators and denominators: (2 · 3)/(3 · 4) Cancel common factor 3: (2)/(4) Simplify by dividing numerator and denominator by 2: (2 ÷ 2)/(4 ÷ 2) = (1)/(2) (1)/(2)