QuestionFind $\frac{8^{x}}{2^{y}}$ if $3x - y = 12$ . Choices: A) $2^{12}$ B) $4^{4}$ C) $8^{2}$ D) Cannot determine.

Studdy Solution

STEP 1

Assumptions1. The given equation is $3x - y =12$ . We need to find the value of $\frac{8^{x}}{^{y}}$

STEP 2

First, we need to express8 and2 in terms of the same base. We know that $8 =2^{}$ , so we can substitute $2^{}$ for8 in the expression $\frac{8^{x}}{2^{y}}$ .
$\frac{8^{x}}{2^{y}} = \frac{(2^{})^{x}}{2^{y}}$

STEP 3

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

STEP 4

Now, we simplify the expression further using the rule of exponents that states $\frac{a^{m}}{a^{n}} = a^{m-n}$ .
$\frac{2^{3x}}{2^{y}} =2^{3x-y}$

STEP 5

We know from the given equation that $3x - y =12$ . So, we substitute $12$ for $3x - y$ in the expression $2^{3x-y}$ .
$2^{3x-y} =2^{12}$ Therefore, the value of $\frac{8^{x}}{2^{y}}$ is $2^{12}$ .

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QuestionFind $\frac{8^{x}}{2^{y}}$ if $3x - y = 12$ . Choices: A) $2^{12}$ B) $4^{4}$ C) $8^{2}$ D) Cannot determine.

Studdy Solution

STEP 1

Assumptions1. The given equation is $3x - y =12$ . We need to find the value of $\frac{8^{x}}{^{y}}$

STEP 2

First, we need to express8 and2 in terms of the same base. We know that $8 =2^{}$ , so we can substitute $2^{}$ for8 in the expression $\frac{8^{x}}{2^{y}}$ .
$\frac{8^{x}}{2^{y}} = \frac{(2^{})^{x}}{2^{y}}$

STEP 3

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

STEP 4

Now, we simplify the expression further using the rule of exponents that states $\frac{a^{m}}{a^{n}} = a^{m-n}$ .
$\frac{2^{3x}}{2^{y}} =2^{3x-y}$

STEP 5

We know from the given equation that $3x - y =12$ . So, we substitute $12$ for $3x - y$ in the expression $2^{3x-y}$ .
$2^{3x-y} =2^{12}$ Therefore, the value of $\frac{8^{x}}{2^{y}}$ is $2^{12}$ .

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QuestionFind 8x2y\frac{8^{x}}{2^{y}}2y8x​ if 3x−y=123x - y = 123x−y=12. Choices: A) 2122^{12}212 B) 444^{4}44 C) 828^{2}82 D) Cannot determine.

Studdy Solution

STEP 1

Assumptions1. The given equation is 3x−y=123x - y =123x−y=12 . We need to find the value of 8xy\frac{8^{x}}{^{y}}y8x​

STEP 2

First, we need to express8 and2 in terms of the same base. We know that 8=28 =2^{}8=2, so we can substitute 22^{}2 for8 in the expression 8x2y\frac{8^{x}}{2^{y}}2y8x​.8x2y=(2)x2y\frac{8^{x}}{2^{y}} = \frac{(2^{})^{x}}{2^{y}}2y8x​=2y(2)x​

STEP 3

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

STEP 4

Now, we simplify the expression further using the rule of exponents that states aman=am−n\frac{a^{m}}{a^{n}} = a^{m-n}anam​=am−n.23x2y=23x−y\frac{2^{3x}}{2^{y}} =2^{3x-y}2y23x​=23x−y

STEP 5

We know from the given equation that 3x−y=123x - y =123x−y=12. So, we substitute 121212 for 3x−y3x - y3x−y in the expression 23x−y2^{3x-y}23x−y.23x−y=2122^{3x-y} =2^{12}23x−y=212Therefore, the value of 8x2y\frac{8^{x}}{2^{y}}2y8x​ is 2122^{12}212.

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QuestionFind $\frac{8^{x}}{2^{y}}$ if $3x - y = 12$ . Choices: A) $2^{12}$ B) $4^{4}$ C) $8^{2}$ D) Cannot determine.

Assumptions1. The given equation is $3x - y =12$ . We need to find the value of $\frac{8^{x}}{^{y}}$

First, we need to express8 and2 in terms of the same base. We know that $8 =2^{}$ , so we can substitute $2^{}$ for8 in the expression $\frac{8^{x}}{2^{y}}$ .
$\frac{8^{x}}{2^{y}} = \frac{(2^{})^{x}}{2^{y}}$

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

Now, we simplify the expression further using the rule of exponents that states $\frac{a^{m}}{a^{n}} = a^{m-n}$ .
$\frac{2^{3x}}{2^{y}} =2^{3x-y}$

We know from the given equation that $3x - y =12$ . So, we substitute $12$ for $3x - y$ in the expression $2^{3x-y}$ .
$2^{3x-y} =2^{12}$ Therefore, the value of $\frac{8^{x}}{2^{y}}$ is $2^{12}$ .