Question

simplify the following expression.

\\(\sqrt{100a^{12}b^8c^2} = \\)

options:

• \\(10a^6b^5c\\)
• \\(10a^6b^4c\\)
• \\(11a^6b^5c\\)
• \\(11a^2bc^5\sqrt{a}\\)

Explanation:

🆕 New Concept Discovered: Simplifying Radical Expressions
How to take square roots of variables with exponents

Step 1: Split the radical

We can break the square root of a product into the product of individual square roots:

$$ \sqrt{100 a^{12} b^{8} c^{2}} = \sqrt{100} \cdot \sqrt{a^{12}} \cdot \sqrt{b^{8}} \cdot \sqrt{c^{2}} $$

Step 2: Simplify the numerical coefficient

Find the square root of the constant number:

$$ \sqrt{100} = 10 $$

Step 3: Simplify the variable terms

To take the square root of a variable with an exponent, we divide its exponent by \(2\) because \(\sqrt{x^n} = x^{\frac{n}{2}}\):

• For \(a^{12}\):

$$ \sqrt{a^{12}} = a^{\frac{12}{2}} = a^{6} $$

• For \(b^{8}\):

$$ \sqrt{b^{8}} = b^{\frac{8}{2}} = b^{4} $$

• For \(c^{2}\):

$$ \sqrt{c^{2}} = c^{\frac{2}{2}} = c^{1} = c $$

Step 4: Combine the simplified parts

Multiply all the simplified components back together:

$$ 10 \cdot a^{6} \cdot b^{4} \cdot c = 10a^{6}b^{4}c $$

Answer:

$$ 10a^{6}b^{4}c $$