Questions: 14/7a^3 = /14a^3b^2 14/7a^3 = square/14a^3b^2

14/7a^3 = /14a^3b^2
14/7a^3 = square/14a^3b^2
Transcript text: \[ \begin{array}{l} \frac{14}{7 a^{3}}=\frac{}{14 a^{3} b^{2}} \\ \frac{14}{7 a^{3}}=\frac{\square}{14 a^{3} b^{2}} \end{array} \]
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Solution

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Solution Steps

To rewrite the given rational expression with a new denominator, we need to determine what factor is missing in the original denominator to make it equal to the new denominator. Once identified, multiply both the numerator and the denominator of the original expression by this factor to obtain the equivalent expression.

To solve the given problem, we need to rewrite the rational expression 147a3\frac{14}{7a^3} as an equivalent rational expression with the denominator 14a3b214a^3b^2.

Step 1: Identify the Original Expression

The original rational expression is: 147a3 \frac{14}{7a^3}

Step 2: Identify the New Denominator

The new denominator we want is: 14a3b2 14a^3b^2

Step 3: Determine the Multiplicative Factor

To convert the original denominator 7a37a^3 to the new denominator 14a3b214a^3b^2, we need to find the factor by which we multiply 7a37a^3 to get 14a3b214a^3b^2.

First, compare the coefficients:

  • The original coefficient is 7, and the new coefficient is 14. Thus, the multiplicative factor for the coefficient is 147=2\frac{14}{7} = 2.

Next, compare the variables:

  • The original denominator has a3a^3, and the new denominator also has a3a^3, so no additional factor is needed for aa.
  • The new denominator has b2b^2, which is not present in the original denominator. Therefore, we need to multiply by b2b^2.

Thus, the overall multiplicative factor is 2b22b^2.

Step 4: Multiply the Numerator and Denominator

Multiply both the numerator and the denominator of the original expression by the factor 2b22b^2 to obtain the equivalent expression: 14×2b27a3×2b2=28b214a3b2 \frac{14 \times 2b^2}{7a^3 \times 2b^2} = \frac{28b^2}{14a^3b^2}

Final Answer

The equivalent rational expression is: 28b214a3b2 \boxed{\frac{28b^2}{14a^3b^2}}

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