Solved on Feb 22, 2024

Find equivalent rational expression with denominator $a^2 b^2 c^2$ for $\frac{8 + 7c}{a^2 b^2 c}$ .

STEP 1

Assumptions
1. The original expression is $\frac{8+7c}{a^{2}b^{2}c}$ .
2. The desired denominator is $a^{2}b^{2}c^{2}$ .
3. The numerator of the equivalent expression must be adjusted to maintain the value of the original expression when the denominator is changed.

STEP 2

Identify the relationship between the original denominator and the desired denominator.
$Original\, denominator = a^{2}b^{2}c$ $Desired\, denominator = a^{2}b^{2}c^{2}$

STEP 3

Determine the factor by which the original denominator must be multiplied to obtain the desired denominator.
$Factor = \frac{Desired\, denominator}{Original\, denominator} = \frac{a^{2}b^{2}c^{2}}{a^{2}b^{2}c}$

STEP 4

Simplify the factor by canceling out common terms in the numerator and denominator.
$Factor = \frac{a^{2}b^{2}c^{2}}{a^{2}b^{2}c} = c$

STEP 5

Multiply the original expression by a fraction equivalent to 1 that has the factor from STEP_4 as both the numerator and the denominator.
$\frac{8+7c}{a^{2}b^{2}c} \times \frac{c}{c}$

STEP 6

Multiply the numerators and the denominators separately.
$\frac{(8+7c) \times c}{a^{2}b^{2}c \times c}$

STEP 7

Simplify the expression by performing the multiplication in the numerator.
$\frac{8c + 7c^2}{a^{2}b^{2}c^2}$

STEP 8

Write the final equivalent rational expression with the given denominator.
$\frac{8+7c}{a^{2}b^{2}c} = \frac{8c + 7c^2}{a^{2}b^{2}c^2}$
The equivalent rational expression with the given denominator $a^{2}b^{2}c^{2}$ is $\frac{8c + 7c^2}{a^{2}b^{2}c^2}$ .

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Find equivalent rational expression with denominator $a^2 b^2 c^2$ for $\frac{8 + 7c}{a^2 b^2 c}$ .

STEP 1

Assumptions
1. The original expression is $\frac{8+7c}{a^{2}b^{2}c}$ .
2. The desired denominator is $a^{2}b^{2}c^{2}$ .
3. The numerator of the equivalent expression must be adjusted to maintain the value of the original expression when the denominator is changed.

STEP 2

Identify the relationship between the original denominator and the desired denominator.
$Original\, denominator = a^{2}b^{2}c$ $Desired\, denominator = a^{2}b^{2}c^{2}$

STEP 3

Determine the factor by which the original denominator must be multiplied to obtain the desired denominator.
$Factor = \frac{Desired\, denominator}{Original\, denominator} = \frac{a^{2}b^{2}c^{2}}{a^{2}b^{2}c}$

STEP 4

Simplify the factor by canceling out common terms in the numerator and denominator.
$Factor = \frac{a^{2}b^{2}c^{2}}{a^{2}b^{2}c} = c$

STEP 5

Multiply the original expression by a fraction equivalent to 1 that has the factor from STEP_4 as both the numerator and the denominator.
$\frac{8+7c}{a^{2}b^{2}c} \times \frac{c}{c}$

STEP 6

Multiply the numerators and the denominators separately.
$\frac{(8+7c) \times c}{a^{2}b^{2}c \times c}$

STEP 7

Simplify the expression by performing the multiplication in the numerator.
$\frac{8c + 7c^2}{a^{2}b^{2}c^2}$

STEP 8

Write the final equivalent rational expression with the given denominator.
$\frac{8+7c}{a^{2}b^{2}c} = \frac{8c + 7c^2}{a^{2}b^{2}c^2}$
The equivalent rational expression with the given denominator $a^{2}b^{2}c^{2}$ is $\frac{8c + 7c^2}{a^{2}b^{2}c^2}$ .

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Find equivalent rational expression with denominator a2b2c2a^2 b^2 c^2a2b2c2 for 8+7ca2b2c\frac{8 + 7c}{a^2 b^2 c}a2b2c8+7c​.

STEP 1

Assumptions1. The original expression is 8+7ca2b2c\frac{8+7c}{a^{2}b^{2}c}a2b2c8+7c​.2. The desired denominator is a2b2c2a^{2}b^{2}c^{2}a2b2c2.3. The numerator of the equivalent expression must be adjusted to maintain the value of the original expression when the denominator is changed.

STEP 2

Identify the relationship between the original denominator and the desired denominator.Original denominator=a2b2cOriginal\, denominator = a^{2}b^{2}cOriginaldenominator=a2b2c Desired denominator=a2b2c2Desired\, denominator = a^{2}b^{2}c^{2}Desireddenominator=a2b2c2

STEP 3

STEP 4

Simplify the factor by canceling out common terms in the numerator and denominator.Factor=a2b2c2a2b2c=cFactor = \frac{a^{2}b^{2}c^{2}}{a^{2}b^{2}c} = cFactor=a2b2ca2b2c2​=c

STEP 5

Multiply the original expression by a fraction equivalent to 1 that has the factor from STEP_4 as both the numerator and the denominator.8+7ca2b2c×cc\frac{8+7c}{a^{2}b^{2}c} \times \frac{c}{c}a2b2c8+7c​×cc​

STEP 6

Multiply the numerators and the denominators separately.(8+7c)×ca2b2c×c\frac{(8+7c) \times c}{a^{2}b^{2}c \times c}a2b2c×c(8+7c)×c​

STEP 7

Simplify the expression by performing the multiplication in the numerator.8c+7c2a2b2c2\frac{8c + 7c^2}{a^{2}b^{2}c^2}a2b2c28c+7c2​

STEP 8

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Find equivalent rational expression with denominator $a^2 b^2 c^2$ for $\frac{8 + 7c}{a^2 b^2 c}$ .

Assumptions
1. The original expression is $\frac{8+7c}{a^{2}b^{2}c}$ .
2. The desired denominator is $a^{2}b^{2}c^{2}$ .
3. The numerator of the equivalent expression must be adjusted to maintain the value of the original expression when the denominator is changed.

Identify the relationship between the original denominator and the desired denominator.
$Original\, denominator = a^{2}b^{2}c$ $Desired\, denominator = a^{2}b^{2}c^{2}$

Simplify the factor by canceling out common terms in the numerator and denominator.
$Factor = \frac{a^{2}b^{2}c^{2}}{a^{2}b^{2}c} = c$

Multiply the original expression by a fraction equivalent to 1 that has the factor from STEP_4 as both the numerator and the denominator.
$\frac{8+7c}{a^{2}b^{2}c} \times \frac{c}{c}$

Multiply the numerators and the denominators separately.
$\frac{(8+7c) \times c}{a^{2}b^{2}c \times c}$

Simplify the expression by performing the multiplication in the numerator.
$\frac{8c + 7c^2}{a^{2}b^{2}c^2}$