Equivalent Expression Analysis: Comparing 45+5a+3ab+27b with Multiple Forms

Q: How do I know which expressions are equivalent without expanding everything?

Look for patterns first! Check if terms have the same coefficients and variables. Then expand step-by-step to verify. Sometimes you can spot obvious differences like missing terms.

Q: Why does option d depend on conditions about b?

Option d has $ 8ab $ instead of $ 3ab + 5a $. These are only equal when $ 8ab = 3ab + 5a $, which means 5ab = 5a, so a must equal 1 or special conditions apply.

Q: What's the easiest way to expand (5+3b)(a+9)?

Use FOIL method:First: 5 × a = 5aOuter: 5 × 9 = 45Inner: 3b × a = 3abLast: 3b × 9 = 27bThen add: $ 5a + 45 + 3ab + 27b $

Q: How do I handle complex fractions like in option c?

Simplify step-by-step! $ \frac{9}{a} \cdot \frac{3ba}{1} = \frac{9 \cdot 3ba}{a \cdot 1} = \frac{27ba}{a} = 27b $ when a ≠ 0. Break complex expressions into smaller parts!

Q: Should I always rearrange terms in the same order?

Yes, it helps with comparison! Arrange terms by degree (highest power first) or alphabetically by variable. This makes it easier to spot equivalent expressions and avoid missing terms.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Choose the expressions equal to the given expression

00:08 Open parentheses properly, multiply each factor by each factor

00:17 This expression appears equal, let's move to the next

00:39 Open parentheses properly, multiply by each factor

00:43 This expression is not equal, let's move to the next

00:54 Let's reduce what we can

01:00 Open parentheses properly, multiply by each factor

01:07 This expression appears equal, let's move to the next

01:25 Let's reduce what we can

01:50 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Which expressions represent the same value?

$45+5a+3ab+27b$

a. $(5+3b)(a+9)$

b. $45+\frac{2}{3}a+4\frac{1}{3}ab+(3a+27)b$

c. $2ab+5(9+a)+ab+\frac{9}{a}\cdot\frac{3ba}{1}$

d. $45+8ab+27b$

2

Step-by-step solution

To solve this problem, we will simplify each given expression and compare the results to the initial expression $45 + 5a + 3ab + 27b$ .

Let's analyze each option:

Option a: $(5+3b)(a+9)$
Distribute: $= 5a + 45 + 3ba + 27b$
Which rearranges to: $45 + 5a + 3ab + 27b$
This matches the original expression.
Option b: $45+\frac{2}{3}a+4\frac{1}{3}ab+(3a+27)b$
Simplify: $= 45 + \frac{2}{3}a + \frac{13}{3}ab + 3ab + 27b$
The terms do not simplify to match the original expression $45 + 5a + 3ab + 27b$ .
Option c: $2ab+5(9+a)+ab+\frac{9}{a}\cdot\frac{3ba}{1}$
Simplify: Complete multiplication and distribution: $= 2ab + 45 + 5a + ab + 27b$
Combine like terms: $= 45 + 5a + 3ab + 27b$
This matches the original expression.
Option d: $45+8ab+27b$
This expression: $= 45 + 8ab + 27b$
Depends on the situation with $b$ , might match if conditions hold, but generally does not match unless specified.

After careful comparison, option a and c definitely represent the same value as the original expression, while option d depends on specific conditions regarding $b$ .

The correct answer is: a, c, d (d. depends on b).

3

Final Answer

a. c. d. (d. depends on b)

Key Points to Remember

Essential concepts to master this topic

Distribution Rule: Multiply each term in first parentheses by each in second
Technique: $(5+3b)(a+9) = 5a + 45 + 3ab + 27b$
Check: Rearrange terms to match original order: $45 + 5a + 3ab + 27b$ ✓

Common Mistakes

Avoid these frequent errors

Only distributing first term in each parentheses
Don't multiply just 5 × a and 3b × 9 = incomplete expansion! This misses cross terms like 5 × 9 and 3b × a. Always multiply every term in the first parentheses by every term in the second parentheses using FOIL or distribution.

Practice Quiz

Test your knowledge with interactive questions

Are the expressions the same or not?

$$ 3+3+3+3 $$

$3\times4$

FAQ

Everything you need to know about this question

How do I know which expressions are equivalent without expanding everything?

+

Look for patterns first! Check if terms have the same coefficients and variables. Then expand step-by-step to verify. Sometimes you can spot obvious differences like missing terms.

Why does option d depend on conditions about b?

+

Option d has $8ab$ instead of $3ab + 5a$ . These are only equal when $8ab = 3ab + 5a$ , which means 5ab = 5a, so a must equal 1 or special conditions apply.

What's the easiest way to expand (5+3b)(a+9)?

+

Use FOIL method:

First: 5 × a = 5a
Outer: 5 × 9 = 45
Inner: 3b × a = 3ab
Last: 3b × 9 = 27b

Then add: $5a + 45 + 3ab + 27b$

How do I handle complex fractions like in option c?

+

Simplify step-by-step! $\frac{9}{a} \cdot \frac{3ba}{1} = \frac{9 \cdot 3ba}{a \cdot 1} = \frac{27ba}{a} = 27b$ when a ≠ 0. Break complex expressions into smaller parts!

Should I always rearrange terms in the same order?

+

Yes, it helps with comparison! Arrange terms by degree (highest power first) or alphabetically by variable. This makes it easier to spot equivalent expressions and avoid missing terms.

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Equivalent Expression Analysis: Comparing 45+5a+3ab+27b with Multiple Forms

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