Rational Expressions | A Comprehensive Guide

Thursday, May 8, 2025

 Rational Expressions: A Comprehensive Guide

1. Introduction 

Imagine you’re baking cookies and need to adjust a recipe. The original recipe calls for (2/x) cups of sugar per batch, but you want to make (x/(x+1)) batches. How much sugar do you need in total?

Objective:

By the end of this lesson, you’ll be able to:

• Simplify, multiply, divide, add, and subtract rational expressions.

• Solve real-world problems involving rational expressions.

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Prerequisite Knowledge Check

Before we start, make sure you know:

1. Factoring polynomials [e.g., $x^2-4=(x+2)(x-2)$]

2. Simplifying fractions (e.g., $\frac{6}{8}=\frac{3}{4}$)

3. Finding the Least Common Denominator (LCD) [e.g., LCD of $\frac{1}{x}$ and $\frac{1}{x+1}$ is $x(x+1)$]

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Core Concept Explanation (I Do – Teacher Models)

What is a Rational Expression?

A rational expression is a fraction where the numerator and denominator are polynomials.

Examples:

• $\frac{3x}{x+2}$

• $\frac{x^2-4}{x^2+5x+6}$

Simplifying Rational Expressions

Step 1: Factor numerator and denominator.

Step 2: Cancel common factors.

Example: Simplify $\frac{x^2-9}{x^2+6x+9}$

1. Factor:

   $$\frac{(x+3)(x-3)}{(x+3{)}^{\mathrm{2}}}$$

2. Cancel $(x+3)$:

   $$\frac{x-3}{x+3}$$
   

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Guided Practice (We Do – Teacher & Students Together)

Problem: Simplify $\frac{2x^2-8}{x^2-x-6}$

Teacher’s Guidance:

• Question: What’s the first step? (Factor numerator and denominator.)

• Question: What factoring techniques apply here? (Difference of squares, trinomial factoring.)

Solution:

1. Factor numerator: $2x^2-8=2(x^2-4)=2(x+2)(x-2)$
   

2. Factor denominator: $x^2-x-6=(x-3)(x+2)$
   

3. Rewrite:

   $$\frac{2(x+2)(x-2)}{(x-3)(x+2)}$$

4. Cancel $(x+2)$:

   $$\frac{2(x-2)}{x-\mathrm{3}}$$

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Independent Practice (You Do – Students Try Alone)

Try these problems on your own before checking the solutions!

1. Simplify $\frac{3x^2-12}{x^2+4x+4}$
   (Hint: Factor the numerator and denominator first.)

2. Simplify $\frac{x^2-5x+6}{x^2-9}$

Solutions:

1. $\frac{3(x-2)}{x+2}$

2. $\frac{x-2}{x+3}$

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Common Mistakes & Troubleshooting

🚨 Watch out for these errors!

• Forgetting to factor completely (e.g., missing common factors).

• Canceling terms, not factors (e.g., $\frac{x+3}{x}$ cannot cancel $x$).

• Ignoring restrictions (denominator cannot be zero).

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Real-World Application

Example: Engineers use rational expressions to calculate resistance in parallel circuits:

$$\frac{1}{R_{total}}=\frac{1}{R_1}+\frac{1}{R_{\mathrm{2}}}$$

Simplifying helps find the total resistance efficiently!

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Summary & Key Takeaways

✔ A rational expression is a fraction of polynomials.
✔ Simplify by factoring and canceling common factors.
✔ Never divide by zero—always state excluded values.

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Practice & Extension

Extra Practice:

1. Simplify $\frac{4x^2-16}{2x^2-8x+8}$

2. Add $\frac{3}{x}+\frac{2}{x+1}$

Challenge Question:
Solve for $x$:

$$\frac{x}{x-2}-\frac{3}{x+2}=\frac{8}{x^2-\mathrm{4}}$$

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Further Resources

📺 Video Tutorials: Inorganic Chemistry Tutor - Simplifying Rational Expressions
📖 Interactive Practice: Desmos Rational Simplifier

Next Lesson: Solving Rational Equations

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More Illustrative Illustrations.

Example 1: Simplify

Expression: $\frac{x^2-4}{x^2+4x+4}$

Solution:

Numerator: $x^2-4=(x+2)(x-2)$

Denominator: $x^2+4x+4=(x+2{)}^2$

Simplified: $\frac{(x+2)(x-2)}{(x+2{)}^2}=\frac{x-2}{x+2}$

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Example 2: Multiply

Expression: $\frac{x+3}{x^2-9}\times \frac{x-3}{x+1}$

Solution:

Factor denominators: $x^2-9=(x+3)(x-3)$

Multiply numerators: $(x+3)(x-3)=x^2-9$

Multiply denominators: $(x+3)(x-3)(x+1)$

Simplified: $\frac{x^2-9}{(x+3)(x-3)(x+1)}=\frac{1}{x+1}$

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Example 3: Divide

Expression: $\frac{2x}{x^2-1}÷\frac{4x^2}{x+1}$

Solution:

Reciprocal and multiply: $\frac{2x}{x^2-1}\times \frac{x+1}{4x^2}$

Factor denominator: $x^2-1=(x+1)(x-1)$

Simplified: $\frac{2x(x+1)}{(x+1)(x-1)(4x^2)}=\frac{1}{2x(x-1)}$

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Example 4: Add

Expression: $\frac{1}{x}+\frac{2}{x+1}$

Solution:

Common denominator: $x(x+1)$

Rewrite fractions: $\frac{x+1}{x(x+1)}+\frac{2x}{x(x+1)}$

Combine numerators: $\frac{x+1+2x}{x(x+1)}=\frac{3x+1}{x(x+1)}$

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Example 5: Subtract

Expression: $\frac{5}{x-2}-\frac{3}{x+2}$

Solution:

Common denominator: $(x-2)(x+2)$

Rewrite fractions: $\frac{5(x+2)}{(x-2)(x+2)}-\frac{3(x-2)}{(x-2)(x+2)}$

Combine numerators: $\frac{5x+10-3x+6}{(x-2)(x+2)}=\frac{2x+16}{(x-2)(x+2)}$

More examples with Partial Solutions. Fill in the blanks what is/are the missing part(s) of the solution.

1. Simplify

Expression: $\frac{x^2-9}{x^2+6x+9}$

Steps:

Numerator:

Denominator: $x^2+6x+9=(\underset{‾}{}{)}^2$

Simplified: ​​​

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2. Multiply

Expression: $\frac{x+2}{x^2-4}\times \frac{x-2}{x+1}$

Steps:

Factor denominator:

Multiply numerators: $(x+2)(x-2)= \underset{}{<a\; href="https://www.blogger.com/comment/fullpage/post/9093914164408134262/7268964984793473683"\; target="\_blank">\_\_\_\_\_\_\_\_\_</a>}$

Result: _________

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3. Divide

Expression: $\frac{3x}{x^2-16}÷\frac{6x^2}{x+4}$

Steps:

Reciprocal: $\frac{3x}{x^2-16}\times$

Factor: _________

Simplified form: _________

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4. Add

Expression: $\frac{2}{x}+\frac{3}{x+2}$

Steps:

Common denominator: ​

Combine: $\frac{2(\underset{‾}{})+3(\underset{‾}{})}{x(x+2)}$

Result: $\frac{\underset{‾}{}}{x(x+2)}$

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5. Subtract

Expression: $\frac{4}{x-3}-\frac{2}{x+1}$

Steps:

Common denominator: ___________​

Combine: $\frac{4(\underset{‾}{})-2(\underset{‾}{})}{(x-3)(x+1)}$

Result: $\frac{\underset{‾}{}}{(x-3)(x+1)}$

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6. Complex Fraction

Expression: $\frac{\frac{x}{x+1\mathrm{<br>}}}{\frac{\mathrm{<br>}\mathrm{<br>}2x}{x^2-1}}$

Steps:

Rewrite: $\frac{x}{x+1}\times \frac{\underset{‾}{}}{\underset{‾}{}}$

Factor:

Simplified: $\frac{\underset{‾}{}}{2}$

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7. Find Restrictions

Expression: $\frac{x^2+5x+6}{x^2-4}$

Steps:

Numerator:

Denominator:

Excluded values: $x\mathrm{\ne \; <a\; href="https://www.blogger.com/comment/fullpage/post/9093914164408134262/7268964984793473683"\; target="\_blank">\_\_\_\_\_\_\_\_\_\_</a>\; and }$$x\mathrm{\ne <a\; href="https://www.blogger.com/comment/fullpage/post/9093914164408134262/7268964984793473683"\; target="\_blank">\_\_\_\_\_\_\_\_\_\_</a>}$

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8. Multiply and Simplify

Expression: $\frac{x^2-1}{x+2}\times \frac{x+2}{x-1}$

Steps:

Factor:

Cancel: $\frac{(x+1)\cancel{(x-1)}\cancel{(x+2)}}{\cancel{(x+2)}\cancel{(x-1)}}$

Result:__________​

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9. Add and Simplify

Expression: $\frac{1}{x-1}+\frac{x}{x^2-1}$

Steps:

Factor: 

Common denominator: _________
$\underline{}$
Result: $\frac{\underset{‾}{}}{(x-1)(x+1)}$

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10. Subtract and Simplify

Expression: $\frac{5}{x^2-x}-\frac{3}{x-1}$

Steps:

Factor:

Common denominator:___________​

Result: $\frac{\underset{‾}{}}{x(x-1)}$

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Answer key

1. $\frac{x-3}{x+3}$

2. $\frac{1}{x+1}$

3. $\frac{1}{2(x-4)}$

4. $\frac{5x+4}{x(x+2)}$

5. $\frac{2x+10}{(x-3)(x+1)}$

6. $\frac{x-1}{2}$

7. $x\mathrm{\ne }2,x\mathrm{\ne }-2$
   

8. $x+1$
   

9. $\frac{2}{x-1}$

10. $\frac{}{}$−3x+5x(x−1)​

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Real-World Rational Expression Problems with Answers

1. Fuel Efficiency

   A car travels $\frac{300}{x+5}$ miles using $\frac{15}{x}$ gallons of gas.
   Simplify the mpg expression.
   Solution: $\frac{20x}{x+5}$ , *(x ≠ 0, -5)*

2. Construction Costs

   Building $\frac{x^2-25}{x-5}$ shelves costs $600.
   Find the cost per shelf.
   Solution: $\frac{600}{x+5}$ (x ≠ 5)

3. Baking Recipe

   A cake needs $\frac{3x}{4}$ cups of flour and $\frac{x}{2}$ cups of sugar.
   Simplify: Flour and sugar ratio.
   Solution: $\frac{3}{2}$ (x ≠ 0)

4. Train Speed

   A train covers $\frac{x^2-36}{x-6}$ km in $\frac{x+6}{3}$ hours.
   Find speed in km/h.
   Solution: 3 (x ≠ 6)

5. Medicine Dosage

   Child's dose is $\frac{4x}{x^2-16}$ mL (adult dose = x mL).
   Simplify dosage.
   Solution: $\frac{4}{x+4}$ , (x ≠ 4, -4)

6. Garden Soil

   Mix $\frac{x+3}{x^2-9}$ lbs of compost with $\frac{1}{x-3}$ lbs of soil.
   Total weight?
   Solution: $\frac{2}{x-3}$ (x ≠ 3, -3)

7. Electricity Bill

   $\frac{x^2-49}{x+7}$ kWh costs $84.
   Cost per kWh?
   Solution: $\frac{84}{x-7}$, (x ≠ -7)

8. Paint Coverage

   $\frac{6x}{x^2-9}$ gallons paints $\frac{x+3}{2}$ walls.
   Gallons per wall?
   Solution: $\frac{12}{x-3}$, (x ≠ 3, -3)

9. Investment Growth

   $$\frac{x-5}{x^2-25}$ grows to $$\frac{1}{x+5}$ in 1 year.
   Growth factor?
   Solution: 1 , (x ≠ 5, -5)

10. Chemistry Lab

    $\frac{x^2-4x+4}{x-2}$ mL acid mixed with $\frac{2x}{5}$ mL water.
    Acid: water ratio?
    Solution: $\frac{5(x-2)}{2x}$ (x ≠ 0, 2)

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Detailed Solution.

​Complete Rational Expressions Practice Worksheet 

1. Fuel Efficiency

Problem:
A car travels $\frac{300}{x+5}$ miles using $\frac{15}{x}$ gallons of gas. Simplify the mpg expression.

Solution:

$$\frac{\frac{300}{x+5}}{\frac{15}{x}}=\frac{300}{x+5}\times \frac{x}{15}=\frac{300x}{\mathrm{15 }(\mathrm{x }+5) }= \frac{20x}{x+5}$$

Restrictions: $x\mathrm{\ne }0,-5$

Answer: $\boxed{{\displaystyle \frac{20x}{x+5}}}$

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2. Construction Costs

Problem:
Building $\frac{x^2-25}{x-5}$ shelves costs $600. Find the cost per shelf.

Solution:

$$\frac{x^2-25}{x-5}=\frac{(x+5)(x-5)}{x-5}=x+5(x\mathrm{\ne }5)$$
$$\text{Cost per shelf}=\frac{600}{x+\mathrm{5}}$$

Answer: $\boxed{{\displaystyle \frac{600}{x+5}}}$

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3. Baking Ratio

Problem:
A recipe uses $\frac{3x}{4}$ cups of flour and $\frac{x}{2}$ cups of sugar. Simplify the flour-to-sugar ratio.

Solution:

$$\frac{\frac{3x}{4}}{\frac{x}{2}}=\frac{3x}{4}\times \frac{2}{x}=\frac{6x}{4x}=\frac{3}{2}$$

Restrictions: $x\mathrm{\ne }0$
Answer: $\boxed{{\displaystyle \frac{3}{2}}}$

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4. Speed Calculation

Problem:
A train covers $\frac{x^2-36}{x-6}$ km in $\frac{x+6}{3}$ hours. Find its speed (km/h).

Solution:

$$\frac{x^2-36}{x-6}=\frac{(x+6)(x-6)}{x-6}=x+6(x\mathrm{\ne }6)$$
$$\text{Speed}=\frac{x+6}{\frac{x+6}{3}}=3$$

Answer: $\boxed{{\displaystyle 3}}$

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5. Medicine Dosage

Problem:
A child’s dose is $\frac{4x}{x^2-16}$ mL (adult dose = $x$ mL). Simplify the expression.

Solution:

$$\frac{4x}{(x+4)(x-4)}=\frac{4}{x+4}(x\mathrm{\ne }4,-4)$$

Answer: $\boxed{{\displaystyle \frac{4}{x+4}}}$

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6. Gardening Mixture

Problem:
Mix $\frac{x+3}{x^2-9}$ lbs of compost with $\frac{1}{x-3}$ lbs of soil. Find the total weight.

Solution:

$$\frac{x+3}{(x+3)(x-3)}+\frac{1}{x-3}=\frac{2}{x-3}(x\mathrm{\ne }3,-3)$$

Answer: $\boxed{{\displaystyle \frac{2}{x-3}}}$

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7. Electricity Cost

Problem:
$\frac{x^2-49}{x+7}$ kWh costs $84. Find the cost per kWh.

Solution:

$$\frac{x^2-49}{x+7}=x-7(x\mathrm{\ne }-7)$$
$$\text{Cost per kWh}=\frac{84}{x-\mathrm{7}}$$

Answer: $\boxed{{\displaystyle \frac{84}{x-7}}}$

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8. Paint Coverage

Problem:
$\frac{6x}{x^2-9}$ gallons covers $\frac{x+3}{2}$ walls. Find gallons per wall.

Solution:

$$\frac{6x}{(x+3)(x-3)}÷\frac{x+3}{2}=\frac{12x}{(x+3{)}^2(x-3)}$$

Restrictions: $x\mathrm{\ne }3,-3$
Answer: $\boxed{{\displaystyle \frac{12}{x^2-9}}}$

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9. Investment Growth

Problem:
An investment grows from $\frac{x-5}{x^2-25}$ to $\frac{1}{x+5}$ dollars. Find the growth factor.

Solution:

$$\frac{x-5}{(x+5)(x-5)}=\frac{1}{x+5}(x\mathrm{\ne }5)$$
$$\text{<br>}$$

Answer: $\boxed{{\displaystyle 1}}$ (no growth)

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10. Chemical Solution Ratio

Problem:
A lab has $\frac{x^2-4x+4}{x-2}$ mL Solution A and $\frac{2x}{5}$ mL Solution B. Find the A : B ratio.

Solution:

$$\frac{(x-2{)}^2}{x-2}=x-2(x\mathrm{\ne }2)<br>$$$$\text{<br>}\text{Ratio}=\frac{x-2}{\frac{2x}{5}}=\frac{5(x-2)}{2x}$$

Answer: $\boxed{{\displaystyle \frac{5(x-2)}{2x}}}$

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Exercises:

Simplify each given problem. The answers are provided with this symbol (→). Just show how the answer ends that way.

1. $\frac{\mathrm{<b>3</b>}\mathrm{<b>x</b>}}{\mathrm{<b>6</b>}{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}}$
   → $\frac{\mathrm{<span\; style="font-size:\; small;">1</span>}}{\mathrm{<span\; style="font-size:\; small;">2</span>}\mathrm{<span\; style="font-size:\; small;">x</span>}}$ (x ≠ 0)

2. $\frac{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>4</b>}}{\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>2</b>}}$
   → $\frac{x-2}{1}$ = x-2, (x ≠ -2)

3. $\frac{\mathrm{<b>5</b>}\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>10</b>}}{\mathrm{<b>15</b>}\mathrm{<b>x</b>}}$
   → $\frac{x+2}{3x}$, (x ≠ 0)

4. $\frac{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>9</b>}}{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>+</b>\mathrm{<b>6</b>}\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>9</b>}}$
   → $\frac{x-3}{x+3}$, (x ≠ -3)

5. $\frac{\mathrm{<b>2</b>}{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>8</b>}\mathrm{<b>x</b>}}{\mathrm{<b>4</b>}\mathrm{<b>x</b>}}$
   → $\frac{x-2}{2}$(x ≠ 0)

6. $\frac{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>5</b>}\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>6</b>}}{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>9</b>}}$
   → $\frac{x-2}{x+3}$ (x ≠ ± 3)

7. $\frac{\mathrm{<b>3</b>}{\mathrm{<b>x</b>}}^{\mathrm{<b>3</b>}}}{\mathrm{<b>9</b>}{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>+</b>\mathrm{<b>6</b>}\mathrm{<b>x</b>}}$
   → $\frac{x^2}{3x+2}$ (x ≠ 0,-⅔)

8. $\frac{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>16</b>}}{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>8</b>}\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>16</b>}}$
   → $\frac{x+4}{x-4}$ (x ≠ 4)

9. $\frac{\mathrm{<b>6</b>}{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}\mathrm{<b>y</b>}}{\mathrm{<b>12</b>}\mathrm{<b>x</b>}{\mathrm{<b>y</b>}}^{\mathrm{<b>2</b>}}}$
   → $\frac{x}{2y}$ , (x ≠ 0, y ≠ 0)

10. $\frac{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>+</b>\mathrm{<b>x</b>}<b>-</b>\mathrm{<b>6</b>}}{{\mathrm{<b>x</b>}}^{\mathrm{<b>2</b>}}<b>-</b>\mathrm{<b>4</b>}\mathrm{<b>x</b>}<b>+</b>\mathrm{<b>4</b>}}$
    → $\frac{\mathrm{<span\; style="font-size:\; x-small;">x</span>}<span\; style="font-size:\; x-small;">+</span>\mathrm{<span\; style="font-size:\; x-small;">3</span>}}{\mathrm{<span\; style="font-size:\; x-small;">x</span>}<span\; style="font-size:\; x-small;">-</span>\mathrm{<span\; style="font-size:\; x-small;">2</span>}}$ , (x ≠ 2)

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Inorganic Chemistry Tutor

Desmos Rational Simplifier

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