Q. \(10 \sqrt{5} \times 10 \sqrt{5}\).

Q: Can I rewrite this as (10\sqrt{5})^2 ?

Yes. (10\sqrt{5})^2 = 10^2\cdot(\sqrt{5})^2 = 100\cdot5 = 500..

Q: Why does \sqrt{5}\cdot\sqrt{5}=5 ?

By definition \sqrt{5} is the positive number whose square is 5, so multiplying \sqrt{5} by itself yields 5.

Q: What general rule lets me multiply expressions like a\sqrt{b}\times c\sqrt{d}?

Multiply coefficients and radicals: a\sqrt{b}\times c\sqrt{d} = (ac)\sqrt{bd}. If bd is a perfect square, simplify further.

Q: Could I simplify before multiplying to make it easier?.

Yes. Combine like parts: 10\sqrt{5}\times10\sqrt{5} = (10\cdot10)(\sqrt{5}\cdot\sqrt{5}) = 100\cdot5 = 500..

Q: What is the decimal value of 10\sqrt{5}\times 10\sqrt{5}?

\sqrt{5} \approx 2.23607 , so 10\sqrt{5} \approx 22.3607 and its square is about 500.0 (exact value 500 ).

Q: How would this change if the radicals were different, e.g., 10\sqrt{5}\times10\sqrt{2}?

Use the rule: 10\sqrt{5}\times10\sqrt{2} = 100\sqrt{10}. That may or may not simplify further depending on 10.

Answer

\[
10\sqrt{5}\times 10\sqrt{5}=100(\sqrt{5})^2=100\cdot5=500.
\]

Detailed Explanation

Write the original expression precisely: \(10\sqrt{5}\times 10\sqrt{5}\).
Use the rule for multiplying products: multiply the numerical coefficients together and multiply the radical factors together. Rewrite the product as the product of coefficients times the product of radicals:
\[
10\sqrt{5}\times 10\sqrt{5} \;=\; (10\cdot 10)\,(\sqrt{5}\cdot\sqrt{5}).
\]
Compute the numerical coefficient product: \(10\cdot 10 = 100\).
Compute the product of the radicals. Since \(\sqrt{5}\cdot\sqrt{5} = (\sqrt{5})^{2} = 5\), we have:
\[
\sqrt{5}\cdot\sqrt{5} = 5.
\]
Combine the two results: \((10\cdot 10)\,(\sqrt{5}\cdot\sqrt{5}) = 100\cdot 5\).
Multiply to obtain the final value: \(100\cdot 5 = 500\).
Therefore, the product is \(500\).

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Algebra FAQs

What is \(10\sqrt{5} \times 10\sqrt{5}\)?

Compute coefficients and radicals: \(10\cdot10\cdot\sqrt{5}\cdot\sqrt{5}=100\cdot5=500..\).

Can I rewrite this as \( (10\sqrt{5})^2 \)?

Yes. \( (10\sqrt{5})^2 = 10^2\cdot(\sqrt{5})^2 = 100\cdot5 = 500..\)

Why does \( \sqrt{5}\cdot\sqrt{5}=5 \)?

By definition \( \sqrt{5} \) is the positive number whose square is 5, so multiplying \( \sqrt{5} \) by itself yields 5.

What general rule lets me multiply expressions like \(a\sqrt{b}\times c\sqrt{d}\)?

Multiply coefficients and radicals: \(a\sqrt{b}\times c\sqrt{d} = (ac)\sqrt{bd}\). If \(bd\) is a perfect square, simplify further.

Could I simplify before multiplying to make it easier?.

Yes. Combine like parts: \(10\sqrt{5}\times10\sqrt{5} = (10\cdot10)(\sqrt{5}\cdot\sqrt{5}) = 100\cdot5 = 500..\)

Is the product always an integer when multiplying identical radical terms?

What is the decimal value of \(10\sqrt{5}\times 10\sqrt{5}\)?

\( \sqrt{5} \approx 2.23607 \), so \( 10\sqrt{5} \approx 22.3607 \) and its square is about \( 500.0 \) (exact value \( 500 \)).

How would this change if the radicals were different, e.g., \(10\sqrt{5}\times10\sqrt{2}\)?

Use the rule: \(10\sqrt{5}\times10\sqrt{2} = 100\sqrt{10}\). That may or may not simplify further depending on \(10\).

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