Solved on Feb 28, 2024

Find the values of a, b, and c in the equation $\frac{9 x^{17} y^{a}}{3 x^{b} y^{2}}=c x^{5} y^{7}$ .

STEP 1

Assumptions
1. We are given the equation $\frac{9 x^{17} y^{a}}{3 x^{b} y^{2}}=c x^{5} y^{7}$ .
2. We need to find the values of $a$ , $b$ , and $c$ .
3. The properties of exponents will be used to solve for the variables.

STEP 2

First, simplify the coefficients and the $x$ terms on the left side of the equation by dividing them.
$\frac{9}{3} = 3$
$x^{17 - b} = x^{17-b}$

STEP 3

Now, simplify the $y$ terms on the left side of the equation by subtracting the exponents.
$y^{a - 2} = y^{a-2}$

STEP 4

After simplification, rewrite the equation with the simplified terms.
$3 x^{17-b} y^{a-2} = c x^{5} y^{7}$

STEP 5

To find the value of $c$ , equate the coefficients on both sides of the equation.
$3 = c$

STEP 6

Now we know that $c = 3$ .

STEP 7

To find the value of $b$ , equate the exponents of $x$ on both sides of the equation.
$17 - b = 5$

STEP 8

Solve for $b$ .
$b = 17 - 5$

STEP 9

Calculate the value of $b$ .
$b = 12$

STEP 10

To find the value of $a$ , equate the exponents of $y$ on both sides of the equation.
$a - 2 = 7$

STEP 11

Solve for $a$ .
$a = 7 + 2$

STEP 12

Calculate the value of $a$ .
$a = 9$
The values of the variables are $a = 9$ , $b = 12$ , and $c = 3$ .

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Find the values of a, b, and c in the equation $\frac{9 x^{17} y^{a}}{3 x^{b} y^{2}}=c x^{5} y^{7}$ .

STEP 1

Assumptions
1. We are given the equation $\frac{9 x^{17} y^{a}}{3 x^{b} y^{2}}=c x^{5} y^{7}$ .
2. We need to find the values of $a$ , $b$ , and $c$ .
3. The properties of exponents will be used to solve for the variables.

STEP 2

First, simplify the coefficients and the $x$ terms on the left side of the equation by dividing them.
$\frac{9}{3} = 3$
$x^{17 - b} = x^{17-b}$

STEP 3

Now, simplify the $y$ terms on the left side of the equation by subtracting the exponents.
$y^{a - 2} = y^{a-2}$

STEP 4

After simplification, rewrite the equation with the simplified terms.
$3 x^{17-b} y^{a-2} = c x^{5} y^{7}$

STEP 5

To find the value of $c$ , equate the coefficients on both sides of the equation.
$3 = c$

STEP 6

Now we know that $c = 3$ .

STEP 7

To find the value of $b$ , equate the exponents of $x$ on both sides of the equation.
$17 - b = 5$

STEP 8

Solve for $b$ .
$b = 17 - 5$

STEP 9

Calculate the value of $b$ .
$b = 12$

STEP 10

To find the value of $a$ , equate the exponents of $y$ on both sides of the equation.
$a - 2 = 7$

STEP 11

Solve for $a$ .
$a = 7 + 2$

STEP 12

Calculate the value of $a$ .
$a = 9$
The values of the variables are $a = 9$ , $b = 12$ , and $c = 3$ .

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Find the values of a, b, and c in the equation 9x17ya3xby2=cx5y7\frac{9 x^{17} y^{a}}{3 x^{b} y^{2}}=c x^{5} y^{7}3xby29x17ya​=cx5y7.

STEP 1

Assumptions1. We are given the equation 9x17ya3xby2=cx5y7\frac{9 x^{17} y^{a}}{3 x^{b} y^{2}}=c x^{5} y^{7}3xby29x17ya​=cx5y7.2. We need to find the values of aaa, bbb, and ccc.3. The properties of exponents will be used to solve for the variables.

STEP 2

First, simplify the coefficients and the xxx terms on the left side of the equation by dividing them.93=3\frac{9}{3} = 339​=3x17−b=x17−bx^{17 - b} = x^{17-b}x17−b=x17−b

STEP 3

Now, simplify the yyy terms on the left side of the equation by subtracting the exponents.ya−2=ya−2y^{a - 2} = y^{a-2}ya−2=ya−2

STEP 4

After simplification, rewrite the equation with the simplified terms.3x17−bya−2=cx5y73 x^{17-b} y^{a-2} = c x^{5} y^{7}3x17−bya−2=cx5y7

STEP 5

To find the value of ccc, equate the coefficients on both sides of the equation.3=c3 = c3=c

STEP 6

Now we know that c=3c = 3c=3.

STEP 7

To find the value of bbb, equate the exponents of xxx on both sides of the equation.17−b=517 - b = 517−b=5

STEP 8

Solve for bbb.b=17−5b = 17 - 5b=17−5

STEP 9

Calculate the value of bbb.b=12b = 12b=12

STEP 10

To find the value of aaa, equate the exponents of yyy on both sides of the equation.a−2=7a - 2 = 7a−2=7

STEP 11

Solve for aaa.a=7+2a = 7 + 2a=7+2

STEP 12

Calculate the value of aaa.a=9a = 9a=9The values of the variables are a=9a = 9a=9, b=12b = 12b=12, and c=3c = 3c=3.

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Find the values of a, b, and c in the equation $\frac{9 x^{17} y^{a}}{3 x^{b} y^{2}}=c x^{5} y^{7}$ .

Assumptions
1. We are given the equation $\frac{9 x^{17} y^{a}}{3 x^{b} y^{2}}=c x^{5} y^{7}$ .
2. We need to find the values of $a$ , $b$ , and $c$ .
3. The properties of exponents will be used to solve for the variables.

First, simplify the coefficients and the $x$ terms on the left side of the equation by dividing them.
$\frac{9}{3} = 3$
$x^{17 - b} = x^{17-b}$

Now, simplify the $y$ terms on the left side of the equation by subtracting the exponents.
$y^{a - 2} = y^{a-2}$

After simplification, rewrite the equation with the simplified terms.
$3 x^{17-b} y^{a-2} = c x^{5} y^{7}$

To find the value of $c$ , equate the coefficients on both sides of the equation.
$3 = c$

Now we know that $c = 3$ .

To find the value of $b$ , equate the exponents of $x$ on both sides of the equation.
$17 - b = 5$

Solve for $b$ .
$b = 17 - 5$

Calculate the value of $b$ .
$b = 12$

To find the value of $a$ , equate the exponents of $y$ on both sides of the equation.
$a - 2 = 7$

Solve for $a$ .
$a = 7 + 2$

Calculate the value of $a$ .
$a = 9$
The values of the variables are $a = 9$ , $b = 12$ , and $c = 3$ .