How to distribute square roots without making common mistakes?

In summary, the conversation discusses a homeschooled student who is struggling with a math problem and seeking help. The student is using a textbook to teach themselves but has no one to explain it to. They realize they made a mistake in the problem and seek clarification on why they need to change all plus signs to minus signs. The expert explains that this is a common mistake and provides a simplified explanation using the distributive property.
  • #1
Cuberoot1
4
0
I'm homeschooled, but it's gotten to the point that my Mom doesn't know how to do what she's teaching me anymore. So now I'm teaching myself with just a textbook and no one to explain it to me. I'm stuck on an issue probably simple, but I still need help. I believe I messed up on the last lines.

Please explain in detail.

#1
(sqrt7+4)(sqrt7-1) = (sqrt7+4)sqrt7-(sqrt7+4)1 = (sqrt7sqrt7+4sqrt7)-(1sqrt7+4•1) = 7+4sqrt7-sqrt7+4 = ?11+3sqrt7?

#2
(Sqrt2x+3)(sqrt2x-3) = (sqrt2x+3)sqrt2x-(sqrt2x+3)3 = (sqrt2xsqrt2x+3sqrt2x)-(3sqrt2x+3•3) = 2x+3sqrt2x-3sqrt2x+9 = ?2x+9?

I can't figure out what to add, and what to subtract on the last part.
 
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  • #2
Cuberoot said:
I'm homeschooled, but it's gotten to the point that my Mom doesn't know how to do what she's teaching me anymore. So now I'm teaching myself with just a textbook and no one to explain it to me. I'm stuck on an issue probably simple, but I still need help. I believe I messed up on the last lines.

Please explain in detail.

#1
\(\displaystyle (\sqrt7+4)(\sqrt7-1) = (\sqrt7+4)\sqrt7-(\sqrt7+4)1 = (\sqrt7\sqrt7+4\sqrt7)-(1\sqrt7+4•1) = 7+4\sqrt7-\sqrt7+4 = ?11+3\sqrt7?\)

You're fine up until you distribute the final bracket. You have +4 and you need -4 (the minus sign means change the sign of every term in the bracket)

#2
\(\displaystyle (\sqrt{2x}+3)(\sqrt{2x}-3) = (\sqrt{2x}+3)\sqrt{2x}-(\sqrt{2x}+3)3 = (\sqrt{2x}\sqrt{2x}+3\sqrt{2x})-(3\sqrt{2x}+3•3) = 2x+3\sqrt{2x}-3\sqrt{2x}+9 = ?2x+9?\)

I can't figure out what to add, and what to subtract on the last part.

Are you familiar with the difference of two squares?

\(\displaystyle (a+b)(a-b) = a^2-b^2\)
 
  • #3
SuperSonic4 said:
You're fine up until you distribute the final bracket. You have +4 and you need -4 (the minus sign means change the sign of every term in the bracket)

Could you explain why I change all + to - ?

I understand the rest. Thanks.
 
  • #4
Hi Cuberoot,

This is a common mistake that people make or overlook when quickly doing problems. Remember that $-(a+b)=-a-b$. You have to distribute the negative to both terms inside the parentheses. Maybe it helps to think of it like this instead. $-(a+b)=-1(a+b)$?

In your problem your second step is correct, but you need to fix this part...

\(\displaystyle (\sqrt7\sqrt7+4\sqrt7){\color{red}-(1\sqrt7+4\cdot1)}\)? How should the part in red but simplified?
 

FAQ: How to distribute square roots without making common mistakes?

What is the process for distributing square roots?

The process for distributing square roots involves breaking down a number inside the square root into its prime factors, and then distributing the square root symbol across those factors. For example, the square root of 18 can be distributed as the square root of 9 times the square root of 2.

Can I distribute a square root over a sum or difference of numbers?

Yes, you can distribute a square root over a sum or difference of numbers, as long as the numbers are being added or subtracted within the square root symbol. For example, the square root of (9 + 16) can be distributed as the square root of 9 plus the square root of 16.

How does distributing square roots affect the answer?

Distributing square roots does not change the value of the expression, but it may simplify the expression by breaking it down into smaller, more manageable parts. This can make it easier to solve or work with the expression.

Is it possible to distribute a square root over a fraction?

Yes, you can distribute a square root over a fraction by distributing the square root symbol to both the numerator and denominator of the fraction. For example, the square root of (9/4) can be distributed as (square root of 9) divided by (square root of 4).

Are there any rules or limitations for distributing square roots?

Yes, there are a few rules to keep in mind when distributing square roots. First, you can only distribute the square root symbol over multiplication or division. Second, you can only distribute the square root symbol over numbers, not variables. Third, you cannot distribute a square root over a sum or difference of numbers if the numbers are being multiplied or divided within the square root symbol.

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