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Simplify the Radical Expression $\\sqrt{9x^{12}}$ (with $x>0$)
Mathematics
Grade 9 (Junior High School)
Question Content
Express in simplest radical form assuming $x > 0$. $\sqrt{9x^{12}}$
Correct Answer
$3x^6$
Detailed Solution Steps
1
Step 1: Use the property of square roots $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ (where $a\\geq0, b\\geq0$) to split the radical: $\sqrt{9x^{12}} = \\sqrt{9} \\times \\sqrt{x^{12}}$
2
Step 2: Calculate $\sqrt{9}$: since $3^2=9$, $\sqrt{9}=3$
3
Step 3: Calculate $\sqrt{x^{12}}$ using the exponent rule $\sqrt{x^n}=x^{\\frac{n}{2}}$ (for $x>0$): $\sqrt{x^{12}}=x^{\\frac{12}{2}}=x^6$
4
Step 4: Multiply the two results together: $3 \\times x^6 = 3x^6$
Knowledge Points Involved
1
Product Property of Square Roots
This property states that $\sqrt{ab} = \sqrt{a} \\times \sqrt{b}$ for non-negative real numbers $a$ and $b$. It allows splitting a square root of a product into the product of square roots, which simplifies radical expressions by breaking them into smaller, easier-to-solve parts.
2
Square Roots of Perfect Squares
A perfect square is a number or expression that is the square of another number or expression. For any non-negative real number $a$, $\sqrt{a^2}=a$. For integers, this means $\sqrt{9}=3$ because $3^2=9$, and for variable expressions, $\sqrt{x^{2n}}=x^n$ when $x>0$.
3
Exponent Rules for Radicals
The rule $\sqrt{x^n}=x^{\\frac{n}{2}}$ (for $x>0$) connects radicals to rational exponents. It works because taking the square root is the inverse operation of raising a number to the 2nd power, so it divides the exponent by 2. This is used to simplify variable expressions inside square roots.
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