Solved on Oct 22, 2023

Find $\frac{8^{x}}{32^{y}}$ given $3x-5y=2$ .

STEP 1

Assumptions1. The equation given is $3x -5y =$ . We need to find the value of $\frac{8^{x}}{32^{y}}$

STEP 2

We first need to express $8$ and $32$ as powers of $2$ . This is because $8$ and $32$ are both powers of $2$ and this will simplify the equation.
$8 =2^{}$ $32 =2^{5}$

STEP 3

Substitute these values into the expression $\frac{8^{x}}{32^{y}}$ .
$\frac{8^{x}}{32^{y}} = \frac{(2^{3})^{x}}{(2^{5})^{y}}$

STEP 4

Using the power of a power rule, we simplify the expression. The power of a power rule states that $(a^{m})^{n} = a^{mn}$ .
$\frac{(2^{3})^{x}}{(2^{})^{y}} = \frac{2^{3x}}{2^{y}}$

STEP 5

We can simplify this further using the quotient of powers rule. The quotient of powers rule states that $\frac{a^{m}}{a^{n}} = a^{m-n}$ .
$\frac{2^{3x}}{2^{5y}} =2^{3x-5y}$

STEP 6

We know from the given equation that $3x -5y =2$ . We can substitute this into the expression.
$2^{3x-5y} =2^{2}$

STEP 7

implify the right side of the equation.
$2^{2} =4$ So, $\frac{^{x}}{32^{y}} =4$ .

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Find $\frac{8^{x}}{32^{y}}$ given $3x-5y=2$ .

STEP 1

Assumptions1. The equation given is $3x -5y =$ . We need to find the value of $\frac{8^{x}}{32^{y}}$

STEP 2

We first need to express $8$ and $32$ as powers of $2$ . This is because $8$ and $32$ are both powers of $2$ and this will simplify the equation.
$8 =2^{}$ $32 =2^{5}$

STEP 3

Substitute these values into the expression $\frac{8^{x}}{32^{y}}$ .
$\frac{8^{x}}{32^{y}} = \frac{(2^{3})^{x}}{(2^{5})^{y}}$

STEP 4

Using the power of a power rule, we simplify the expression. The power of a power rule states that $(a^{m})^{n} = a^{mn}$ .
$\frac{(2^{3})^{x}}{(2^{})^{y}} = \frac{2^{3x}}{2^{y}}$

STEP 5

We can simplify this further using the quotient of powers rule. The quotient of powers rule states that $\frac{a^{m}}{a^{n}} = a^{m-n}$ .
$\frac{2^{3x}}{2^{5y}} =2^{3x-5y}$

STEP 6

We know from the given equation that $3x -5y =2$ . We can substitute this into the expression.
$2^{3x-5y} =2^{2}$

STEP 7

implify the right side of the equation.
$2^{2} =4$ So, $\frac{^{x}}{32^{y}} =4$ .

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Find 8x32y\frac{8^{x}}{32^{y}}32y8x​ given 3x−5y=23x-5y=23x−5y=2.

STEP 1

Assumptions1. The equation given is 3x−5y=3x -5y =3x−5y= . We need to find the value of 8x32y\frac{8^{x}}{32^{y}}32y8x​

STEP 2

We first need to express 888 and 323232 as powers of 222. This is because 888 and 323232 are both powers of 222 and this will simplify the equation.8=28 =2^{}8=232=2532 =2^{5}32=25

STEP 3

Substitute these values into the expression 8x32y\frac{8^{x}}{32^{y}}32y8x​.8x32y=(23)x(25)y\frac{8^{x}}{32^{y}} = \frac{(2^{3})^{x}}{(2^{5})^{y}}32y8x​=(25)y(23)x​

STEP 4

Using the power of a power rule, we simplify the expression. The power of a power rule states that (am)n=amn(a^{m})^{n} = a^{mn}(am)n=amn.(23)x(2)y=23x2y\frac{(2^{3})^{x}}{(2^{})^{y}} = \frac{2^{3x}}{2^{y}}(2)y(23)x​=2y23x​

STEP 5

We can simplify this further using the quotient of powers rule. The quotient of powers rule states that aman=am−n\frac{a^{m}}{a^{n}} = a^{m-n}anam​=am−n.23x25y=23x−5y\frac{2^{3x}}{2^{5y}} =2^{3x-5y}25y23x​=23x−5y

STEP 6

We know from the given equation that 3x−5y=23x -5y =23x−5y=2. We can substitute this into the expression.23x−5y=222^{3x-5y} =2^{2}23x−5y=22

STEP 7

implify the right side of the equation.22=42^{2} =422=4So, x32y=4\frac{^{x}}{32^{y}} =432yx​=4.

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Find $\frac{8^{x}}{32^{y}}$ given $3x-5y=2$ .

Assumptions1. The equation given is $3x -5y =$ . We need to find the value of $\frac{8^{x}}{32^{y}}$

We first need to express $8$ and $32$ as powers of $2$ . This is because $8$ and $32$ are both powers of $2$ and this will simplify the equation.
$8 =2^{}$ $32 =2^{5}$

Substitute these values into the expression $\frac{8^{x}}{32^{y}}$ .
$\frac{8^{x}}{32^{y}} = \frac{(2^{3})^{x}}{(2^{5})^{y}}$

Using the power of a power rule, we simplify the expression. The power of a power rule states that $(a^{m})^{n} = a^{mn}$ .
$\frac{(2^{3})^{x}}{(2^{})^{y}} = \frac{2^{3x}}{2^{y}}$

We can simplify this further using the quotient of powers rule. The quotient of powers rule states that $\frac{a^{m}}{a^{n}} = a^{m-n}$ .
$\frac{2^{3x}}{2^{5y}} =2^{3x-5y}$

We know from the given equation that $3x -5y =2$ . We can substitute this into the expression.
$2^{3x-5y} =2^{2}$

implify the right side of the equation.
$2^{2} =4$ So, $\frac{^{x}}{32^{y}} =4$ .