Expand (7b+3z)²: Converting Between Power and Sum Notations

Q: How do I remember the perfect square formula?

Think "First squared, twice the product, last squared". For $(7b+3z)^2$: first term squared is $49b^2$, twice the product is $42bz$, last squared is $9z^2$.

Q: Can I use FOIL instead of the perfect square formula?

Yes! FOIL gives the same result: First (7b×7b), Outer (7b×3z), Inner (3z×7b), Last (3z×3z). You'll get $49b^2 + 21bz + 21bz + 9z^2 = 49b^2 + 42bz + 9z^2$.

Q: What's the difference between a sum and a power in this problem?

The sum form shows all terms added together: $49b^2 + 42bz + 9z^2$. The power form shows the original expression with an exponent: $(7b+3z)^2$. They're equal!

Q: How do I check if my expansion is correct?

Use FOIL to multiply $(7b+3z)(7b+3z)$ step by step, or substitute simple values like b=1, z=1 into both forms. If $(7+3)^2 = 100$ and $49+42+9 = 100$, you're right!

Binomial Expansion with Perfect Square Forms

Express the following exercise as a sum and as a power:

$(7b+3z)(7b+3z)=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express as sum and power

00:04 Double factor itself is actually squared

00:14 Use this formula and square it

00:22 Use shortened multiplication formulas to open the parentheses

00:36 When 7B is A

00:39 And 3Z is B

00:44 We'll substitute according to the formula and solve

00:54 Solve the squares and multiplications

01:06 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Express the following exercise as a sum and as a power:

$(7b+3z)(7b+3z)=\text{?}$

Step-by-step solution

To express the given expression $(7b+3z)(7b+3z)$ as a sum and a power, we will follow these steps:

Step 1: Identify the components as a binomial expansion:
Let $a = 7b$ and $b = 3z$ .
Step 2: Use the formula for the square of a binomial, which is $(a+b)^2 = a^2 + 2ab + b^2$ .
Step 3: Calculate each term:

$a^2 = (7b)^2 = 49b^2$
$b^2 = (3z)^2 = 9z^2$
$2ab = 2(7b)(3z) = 42bz$

By substituting these into the formula, we get:
$(7b+3z)^2 = 49b^2 + 2(7b)(3z) + 9z^2$

Therefore, the expression as a sum is $49b^2 + 42bz + 9z^2$ , and as a power, it is $(7b+3z)^2$ .

Thus, the solution to the problem is:

$49b^2 + 42bz + 9z^2$

$(7b+3z)^2$

Final Answer

$49b^2+42bz+9z^2$

$(7b+3z)^2$

Key Points to Remember

Essential concepts to master this topic

Perfect Square Formula: $(a+b)^2 = a^2 + 2ab + b^2$ always has three terms
Cross Term Calculation: Middle term is $2(7b)(3z) = 42bz$
Verification Check: Multiply binomials using FOIL: First + Outer + Inner + Last ✓

Common Mistakes

Avoid these frequent errors

Forgetting the middle term in perfect square expansion
Don't expand $(7b+3z)^2$ as just $49b^2 + 9z^2$ = missing the cross term! This ignores the 2ab part of the formula and gives an incomplete answer. Always remember the formula $(a+b)^2 = a^2 + 2ab + b^2$ has three terms.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:

$$ (x+y)^2 $$

FAQ

Everything you need to know about this question

Why is there a middle term 42bz when squaring (7b+3z)?

When you square a binomial, you're multiplying it by itself: $(7b+3z)(7b+3z)$ . The middle term comes from multiplying the different parts: 7b × 3z + 3z × 7b = 42bz. This is the 2ab part of the formula!

How do I remember the perfect square formula?

Think "First squared, twice the product, last squared". For $(7b+3z)^2$ : first term squared is $49b^2$ , twice the product is $42bz$ , last squared is $9z^2$ .

Can I use FOIL instead of the perfect square formula?

Yes! FOIL gives the same result: First (7b×7b), Outer (7b×3z), Inner (3z×7b), Last (3z×3z). You'll get $49b^2 + 21bz + 21bz + 9z^2 = 49b^2 + 42bz + 9z^2$ .

What's the difference between a sum and a power in this problem?

The sum form shows all terms added together: $49b^2 + 42bz + 9z^2$ . The power form shows the original expression with an exponent: $(7b+3z)^2$ . They're equal!

How do I check if my expansion is correct?

Use FOIL to multiply $(7b+3z)(7b+3z)$ step by step, or substitute simple values like b=1, z=1 into both forms. If $(7+3)^2 = 100$ and $49+42+9 = 100$ , you're right!

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