Evaluate ((2/3)^5(2/3)^0(81/16)^-2(2/3)^-1)/((3/2)^-5(2/3)^5*((2/3)^5)^2)

$\frac{{(\frac{2}{3})}^{5} \cdot {(\frac{2}{3})}^{0} \cdot {(\frac{81}{16})}^{- 2} \cdot {(\frac{2}{3})}^{- 1}}{{(\frac{3}{2})}^{- 5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2}}$

Step 1

Multiply ${(\frac{2}{3})}^{5}$ by ${(\frac{2}{3})}^{0}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(\frac{2}{3})}^{5 + 0} \cdot {(\frac{81}{16})}^{- 2} \cdot {(\frac{2}{3})}^{- 1}}{{(\frac{3}{2})}^{- 5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2}}$

Add $5$ and $0$ .

$\frac{{(\frac{2}{3})}^{5} \cdot {(\frac{81}{16})}^{- 2} \cdot {(\frac{2}{3})}^{- 1}}{{(\frac{3}{2})}^{- 5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2}}$

Step 2

Multiply ${(\frac{2}{3})}^{5}$ by ${(\frac{2}{3})}^{- 1}$ by adding the exponents.

Move ${(\frac{2}{3})}^{- 1}$ .

$\frac{{(\frac{2}{3})}^{- 1} {(\frac{2}{3})}^{5} \cdot {(\frac{81}{16})}^{- 2}}{{(\frac{3}{2})}^{- 5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(\frac{2}{3})}^{- 1 + 5} \cdot {(\frac{81}{16})}^{- 2}}{{(\frac{3}{2})}^{- 5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2}}$

Add $- 1$ and $5$ .

$\frac{{(\frac{2}{3})}^{4} \cdot {(\frac{81}{16})}^{- 2}}{{(\frac{3}{2})}^{- 5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2}}$

Step 3

Move ${(\frac{81}{16})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(\frac{2}{3})}^{4}}{{(\frac{3}{2})}^{- 5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2}}$

Step 4

Simplify the denominator.

Rewrite $\frac{3}{2}$ as ${(\frac{2}{3})}^{- 1}$ .

$\frac{{(\frac{2}{3})}^{4}}{{({(\frac{2}{3})}^{- 1})}^{- 5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2}}$

Rewrite $\frac{2}{3}$ as ${(\frac{2}{3})}^{1}$ .

$\frac{{(\frac{2}{3})}^{4}}{{({({(\frac{2}{3})}^{1})}^{- 1})}^{- 5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(\frac{2}{3})}^{4}}{{({(\frac{2}{3})}^{1 \cdot - 1})}^{- 5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2}}$

Multiply $- 1$ by $1$ .

$\frac{{(\frac{2}{3})}^{4}}{{({(\frac{2}{3})}^{- 1})}^{- 5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2}}$

Multiply the exponents in ${({(\frac{2}{3})}^{- 1})}^{- 5}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(\frac{2}{3})}^{4}}{{(\frac{2}{3})}^{- - 5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2}}$

Multiply $- 1$ by $- 5$ .

$\frac{{(\frac{2}{3})}^{4}}{{(\frac{2}{3})}^{5} \cdot {(\frac{2}{3})}^{5} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(\frac{2}{3})}^{4}}{{(\frac{2}{3})}^{5 + 5} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2}}$

Add $5$ and $5$ .

$\frac{{(\frac{2}{3})}^{4}}{{(\frac{2}{3})}^{10} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2}}$

Step 5

Cancel the common factor of ${(\frac{2}{3})}^{4}$ and ${(\frac{2}{3})}^{10}$ .

Multiply by $1$ .

$\frac{{(\frac{2}{3})}^{4} \cdot 1}{{(\frac{2}{3})}^{10} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2}}$

Cancel the common factors.

Factor ${(\frac{2}{3})}^{4}$ out of ${(\frac{2}{3})}^{10} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2}$ .

$\frac{{(\frac{2}{3})}^{4} \cdot 1}{{(\frac{2}{3})}^{4} ({(\frac{2}{3})}^{6} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2})}$

Cancel the common factor.

$\frac{{(\frac{2}{3})}^{4} \cdot 1}{{(\frac{2}{3})}^{4} ({(\frac{2}{3})}^{6} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2})}$

Rewrite the expression.

$\frac{1}{{(\frac{2}{3})}^{6} \cdot {({(\frac{2}{3})}^{5})}^{2} {(\frac{81}{16})}^{2}}$

Step 6

Simplify the denominator.

Rewrite $\frac{81}{16}$ as ${(\frac{16}{81})}^{- 1}$ .

$\frac{1}{{({(\frac{2}{3})}^{5})}^{2} ({(\frac{2}{3})}^{6} {({(\frac{16}{81})}^{- 1})}^{2})}$

Rewrite $\frac{16}{81}$ as ${(\frac{2}{3})}^{4}$ .

$\frac{1}{{({(\frac{2}{3})}^{5})}^{2} ({(\frac{2}{3})}^{6} {({({(\frac{2}{3})}^{4})}^{- 1})}^{2})}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{{({(\frac{2}{3})}^{5})}^{2} ({(\frac{2}{3})}^{6} {({(\frac{2}{3})}^{4 \cdot - 1})}^{2})}$

Multiply $4$ by $- 1$ .

$\frac{1}{{({(\frac{2}{3})}^{5})}^{2} ({(\frac{2}{3})}^{6} {({(\frac{2}{3})}^{- 4})}^{2})}$

Multiply the exponents in ${({(\frac{2}{3})}^{- 4})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{{({(\frac{2}{3})}^{5})}^{2} ({(\frac{2}{3})}^{6} {(\frac{2}{3})}^{- 4 \cdot 2})}$

Multiply $- 4$ by $2$ .

$\frac{1}{{({(\frac{2}{3})}^{5})}^{2} ({(\frac{2}{3})}^{6} {(\frac{2}{3})}^{- 8})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{{({(\frac{2}{3})}^{5})}^{2} {(\frac{2}{3})}^{6 - 8}}$

Subtract $8$ from $6$ .

$\frac{1}{{({(\frac{2}{3})}^{5})}^{2} {(\frac{2}{3})}^{- 2}}$

Step 7

Simplify the denominator.

Apply the product rule to $\frac{2}{3}$ .

$\frac{1}{{(\frac{2^{5}}{3^{5}})}^{2} {(\frac{2}{3})}^{- 2}}$

Apply the product rule to $\frac{2^{5}}{3^{5}}$ .

$\frac{1}{\frac{{(2^{5})}^{2}}{{(3^{5})}^{2}} {(\frac{2}{3})}^{- 2}}$

Apply the product rule to $\frac{2}{3}$ .

$\frac{1}{\frac{{(2^{5})}^{2}}{{(3^{5})}^{2}} \cdot \frac{2^{- 2}}{3^{- 2}}}$

Simplify the numerator.

Multiply the exponents in ${(2^{5})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{\frac{2^{5 \cdot 2}}{{(3^{5})}^{2}} \cdot \frac{2^{- 2}}{3^{- 2}}}$

Multiply $5$ by $2$ .

$\frac{1}{\frac{2^{10}}{{(3^{5})}^{2}} \cdot \frac{2^{- 2}}{3^{- 2}}}$

Raise $2$ to the power of $10$ .

$\frac{1}{\frac{1024}{{(3^{5})}^{2}} \cdot \frac{2^{- 2}}{3^{- 2}}}$

Simplify the denominator.

Multiply the exponents in ${(3^{5})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{\frac{1024}{3^{5 \cdot 2}} \cdot \frac{2^{- 2}}{3^{- 2}}}$

Multiply $5$ by $2$ .

$\frac{1}{\frac{1024}{3^{10}} \cdot \frac{2^{- 2}}{3^{- 2}}}$

Raise $3$ to the power of $10$ .

$\frac{1}{\frac{1024}{59049} \cdot \frac{2^{- 2}}{3^{- 2}}}$

Move $2^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{\frac{1024}{59049} \cdot \frac{1}{3^{- 2} \cdot 2^{2}}}$

Move $3^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{1}{\frac{1024}{59049} \cdot \frac{3^{2}}{2^{2}}}$

Raise $3$ to the power of $2$ .

$\frac{1}{\frac{1024}{59049} \cdot \frac{9}{2^{2}}}$

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

$\frac{1}{\frac{1024}{59049} \cdot \frac{9}{4}}$

Step 8

Simplify terms.

Multiply $\frac{1024}{59049}$ by $\frac{9}{4}$ .

$\frac{1}{\frac{1024 \cdot 9}{59049 \cdot 4}}$

Multiply.

Multiply $1024$ by $9$ .

$\frac{1}{\frac{9216}{59049 \cdot 4}}$

Multiply $59049$ by $4$ .

$\frac{1}{\frac{9216}{236196}}$

Cancel the common factor of $9216$ and $236196$ .

Factor $36$ out of $9216$ .

$\frac{1}{\frac{36 (256)}{236196}}$

Cancel the common factors.

Factor $36$ out of $236196$ .

$\frac{1}{\frac{36 \cdot 256}{36 \cdot 6561}}$

Cancel the common factor.

$\frac{1}{\frac{36 \cdot 256}{36 \cdot 6561}}$

Rewrite the expression.

$\frac{1}{\frac{256}{6561}}$

Step 9

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{6561}{256})$

Step 10

Multiply $\frac{6561}{256}$ by $1$ .

$\frac{6561}{256}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{6561}{256}$

Decimal Form:

$25.62890625$

Mixed Number Form:

$25 \frac{161}{256}$

Evaluate ((2/3)^5*(2/3)^0*(81/16)^-2*(2/3)^-1)/((3/2)^-5*(2/3)^5*((2/3)^5)^2)

Related Questions

Evaluate ((2/3)^5(2/3)^0(81/16)^-2(2/3)^-1)/((3/2)^-5(2/3)^5*((2/3)^5)^2)