Question Rewrite $\left(\frac{11}{9}\right)^{s}$ as a quotient of powers.

Studdy Solution

STEP 1

Assumptions
1. We are given the expression $\left(\frac{11}{9}\right)^{s}$ .
2. We need to express it as a quotient of powers.
3. The variable $s$ is an exponent applied to both the numerator and the denominator.
4. The options provided are in the form of powers of integers or a quotient of powers.

STEP 2

Recall the property of exponents that states when a fraction is raised to a power, the exponent applies to both the numerator and the denominator.
$\left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}$

STEP 3

Apply this property to the given expression $\left(\frac{11}{9}\right)^{s}$ .
$\left(\frac{11}{9}\right)^{s} = \frac{11^{s}}{9^{s}}$

STEP 4

To match the given options, we need to determine the value of $s$ that makes the expression a quotient of powers where the powers are integers.

STEP 5

Looking at the options, we see that the powers of 11 and 9 are 5 and 3, respectively, in option D. This suggests that $s$ could be 5 because the power of 11 is 5, and we can express $9^{5}$ as $9^{3} \cdot 9^{2}$ .

STEP 6

Rewrite $9^{5}$ as a product of $9^{3}$ and $9^{2}$ .
$9^{5} = 9^{3} \cdot 9^{2}$

STEP 7

Now, let's express the denominator $9^{s}$ with $s$ as 5 in terms of $9^{3}$ .
$9^{5} = (9^{3})^{(5/3)}$

STEP 8

Since we cannot have fractional exponents to match the options, we need to find an integer value for $s$ that allows us to express $9^{s}$ as $9^{3}$ .

STEP 9

The only integer value for $s$ that allows this is $s = 3$ , as $9^{3}$ is already in the desired form.

STEP 10

Now that we have determined $s = 3$ , we can substitute this value into the expression $\frac{11^{s}}{9^{s}}$ .
$\frac{11^{s}}{9^{s}} = \frac{11^{3}}{9^{3}}$

STEP 11

We see that option D has the form $\frac{11^{5}}{9^{3}}$ . To match this, we need to express $11^{3}$ in a way that resembles $11^{5}$ .

STEP 12

Notice that we cannot directly convert $11^{3}$ into $11^{5}$ , but we can express $11^{5}$ as $11^{3} \cdot 11^{2}$ .

STEP 13

Since we cannot have additional factors of 11 in the numerator to match the options, we conclude that the value of $s$ must be 3 to match the form of the options, specifically option D.

STEP 14

Therefore, the correct representation of $\left(\frac{11}{9}\right)^{s}$ as a quotient of powers, with $s = 3$ , is option D.
$\left(\frac{11}{9}\right)^{s} = \frac{11^{s}}{9^{s}} = \frac{11^{3}}{9^{3}}$
The correct answer is D. $\frac{11^{5}}{9^{3}}$ .

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Question Rewrite $\left(\frac{11}{9}\right)^{s}$ as a quotient of powers.

Studdy Solution

STEP 1

Assumptions
1. We are given the expression $\left(\frac{11}{9}\right)^{s}$ .
2. We need to express it as a quotient of powers.
3. The variable $s$ is an exponent applied to both the numerator and the denominator.
4. The options provided are in the form of powers of integers or a quotient of powers.

STEP 2

Recall the property of exponents that states when a fraction is raised to a power, the exponent applies to both the numerator and the denominator.
$\left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}$

STEP 3

Apply this property to the given expression $\left(\frac{11}{9}\right)^{s}$ .
$\left(\frac{11}{9}\right)^{s} = \frac{11^{s}}{9^{s}}$

STEP 4

To match the given options, we need to determine the value of $s$ that makes the expression a quotient of powers where the powers are integers.

STEP 5

Looking at the options, we see that the powers of 11 and 9 are 5 and 3, respectively, in option D. This suggests that $s$ could be 5 because the power of 11 is 5, and we can express $9^{5}$ as $9^{3} \cdot 9^{2}$ .

STEP 6

Rewrite $9^{5}$ as a product of $9^{3}$ and $9^{2}$ .
$9^{5} = 9^{3} \cdot 9^{2}$

STEP 7

Now, let's express the denominator $9^{s}$ with $s$ as 5 in terms of $9^{3}$ .
$9^{5} = (9^{3})^{(5/3)}$

STEP 8

Since we cannot have fractional exponents to match the options, we need to find an integer value for $s$ that allows us to express $9^{s}$ as $9^{3}$ .

STEP 9

The only integer value for $s$ that allows this is $s = 3$ , as $9^{3}$ is already in the desired form.

STEP 10

Now that we have determined $s = 3$ , we can substitute this value into the expression $\frac{11^{s}}{9^{s}}$ .
$\frac{11^{s}}{9^{s}} = \frac{11^{3}}{9^{3}}$

STEP 11

We see that option D has the form $\frac{11^{5}}{9^{3}}$ . To match this, we need to express $11^{3}$ in a way that resembles $11^{5}$ .

STEP 12

Notice that we cannot directly convert $11^{3}$ into $11^{5}$ , but we can express $11^{5}$ as $11^{3} \cdot 11^{2}$ .

STEP 13

Since we cannot have additional factors of 11 in the numerator to match the options, we conclude that the value of $s$ must be 3 to match the form of the options, specifically option D.

STEP 14

Therefore, the correct representation of $\left(\frac{11}{9}\right)^{s}$ as a quotient of powers, with $s = 3$ , is option D.
$\left(\frac{11}{9}\right)^{s} = \frac{11^{s}}{9^{s}} = \frac{11^{3}}{9^{3}}$
The correct answer is D. $\frac{11^{5}}{9^{3}}$ .

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Question Rewrite (119)s\left(\frac{11}{9}\right)^{s}(911​)s as a quotient of powers.

Studdy Solution

STEP 1

STEP 2

Recall the property of exponents that states when a fraction is raised to a power, the exponent applies to both the numerator and the denominator.(ab)n=anbn\left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}(ba​)n=bnan​

STEP 3

Apply this property to the given expression (119)s\left(\frac{11}{9}\right)^{s}(911​)s.(119)s=11s9s\left(\frac{11}{9}\right)^{s} = \frac{11^{s}}{9^{s}}(911​)s=9s11s​

STEP 4

To match the given options, we need to determine the value of sss that makes the expression a quotient of powers where the powers are integers.

STEP 5

Looking at the options, we see that the powers of 11 and 9 are 5 and 3, respectively, in option D. This suggests that sss could be 5 because the power of 11 is 5, and we can express 959^{5}95 as 93⋅929^{3} \cdot 9^{2}93⋅92.

STEP 6

Rewrite 959^{5}95 as a product of 939^{3}93 and 929^{2}92.95=93⋅929^{5} = 9^{3} \cdot 9^{2}95=93⋅92

STEP 7

Now, let's express the denominator 9s9^{s}9s with sss as 5 in terms of 939^{3}93.95=(93)(5/3)9^{5} = (9^{3})^{(5/3)}95=(93)(5/3)

STEP 8

Since we cannot have fractional exponents to match the options, we need to find an integer value for sss that allows us to express 9s9^{s}9s as 939^{3}93.

STEP 9

The only integer value for sss that allows this is s=3s = 3s=3, as 939^{3}93 is already in the desired form.

STEP 10

Now that we have determined s=3s = 3s=3, we can substitute this value into the expression 11s9s\frac{11^{s}}{9^{s}}9s11s​.11s9s=11393\frac{11^{s}}{9^{s}} = \frac{11^{3}}{9^{3}}9s11s​=93113​

STEP 11

We see that option D has the form 11593\frac{11^{5}}{9^{3}}93115​. To match this, we need to express 11311^{3}113 in a way that resembles 11511^{5}115.

STEP 12

Notice that we cannot directly convert 11311^{3}113 into 11511^{5}115, but we can express 11511^{5}115 as 113⋅11211^{3} \cdot 11^{2}113⋅112.

STEP 13

Since we cannot have additional factors of 11 in the numerator to match the options, we conclude that the value of sss must be 3 to match the form of the options, specifically option D.

STEP 14

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Question Rewrite $\left(\frac{11}{9}\right)^{s}$ as a quotient of powers.

Recall the property of exponents that states when a fraction is raised to a power, the exponent applies to both the numerator and the denominator.
$\left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}$

Apply this property to the given expression $\left(\frac{11}{9}\right)^{s}$ .
$\left(\frac{11}{9}\right)^{s} = \frac{11^{s}}{9^{s}}$

To match the given options, we need to determine the value of $s$ that makes the expression a quotient of powers where the powers are integers.

Looking at the options, we see that the powers of 11 and 9 are 5 and 3, respectively, in option D. This suggests that $s$ could be 5 because the power of 11 is 5, and we can express $9^{5}$ as $9^{3} \cdot 9^{2}$ .

Rewrite $9^{5}$ as a product of $9^{3}$ and $9^{2}$ .
$9^{5} = 9^{3} \cdot 9^{2}$

Now, let's express the denominator $9^{s}$ with $s$ as 5 in terms of $9^{3}$ .
$9^{5} = (9^{3})^{(5/3)}$

Since we cannot have fractional exponents to match the options, we need to find an integer value for $s$ that allows us to express $9^{s}$ as $9^{3}$ .

The only integer value for $s$ that allows this is $s = 3$ , as $9^{3}$ is already in the desired form.

Now that we have determined $s = 3$ , we can substitute this value into the expression $\frac{11^{s}}{9^{s}}$ .
$\frac{11^{s}}{9^{s}} = \frac{11^{3}}{9^{3}}$

We see that option D has the form $\frac{11^{5}}{9^{3}}$ . To match this, we need to express $11^{3}$ in a way that resembles $11^{5}$ .

Notice that we cannot directly convert $11^{3}$ into $11^{5}$ , but we can express $11^{5}$ as $11^{3} \cdot 11^{2}$ .

Since we cannot have additional factors of 11 in the numerator to match the options, we conclude that the value of $s$ must be 3 to match the form of the options, specifically option D.