Example Card: Simplifying Before Combining
Simplify each radical expression fully before adding or subtracting like radicals. Work through Grade 9 radical simplification step by step.
Key Concepts
Sometimes radicals are in disguise. Let's unmask them to see if they're actually alike. The second key idea is to simplify first, which is often necessary before you can combine terms.
Example Problem Simplify $5\sqrt{12k^3} + 3\sqrt{3k} + 7\sqrt{3k}$, where $k$ is a non negative real number.
Step by Step 1. We examine the expression and see that the first radical, $\sqrt{12k^3}$, might be simplified. The other two, $\sqrt{3k}$, are already in simplest form. $$ 5\sqrt{12k^3} + 3\sqrt{3k} + 7\sqrt{3k} $$ 2. Factor the first radicand, $12k^3$, looking for perfect squares. $$ = 5\sqrt{4 \cdot k^2 \cdot 3k} + 3\sqrt{3k} + 7\sqrt{3k} $$ 3. Apply the Product Property of Radicals to separate the factors. $$ = 5 \cdot \sqrt{4} \cdot \sqrt{k^2} \cdot \sqrt{3k} + 3\sqrt{3k} + 7\sqrt{3k} $$ 4. Simplify the perfect square roots. Remember that $5 \cdot \sqrt{4} \cdot \sqrt{k^2}$ becomes $5 \cdot 2 \cdot k$. $$ = 10k\sqrt{3k} + 3\sqrt{3k} + 7\sqrt{3k} $$ 5. Now all terms are like radicals. Factor out the common radical, $\sqrt{3k}$. $$ = (10k + 3 + 7)\sqrt{3k} $$ 6. Simplify the expression inside the parentheses. $$ = (10k + 10)\sqrt{3k} $$.
Common Questions
What is Simplifying Before Combining in Grade 9 algebra?
It is a core concept in Grade 9 algebra that builds problem-solving skills and prepares students for advanced math coursework.
How do you apply simplifying before combining to solve problems?
Identify the relevant formula or property, substitute known values carefully, apply each step in order, and verify the result makes sense.
What common errors occur with simplifying before combining?
Misapplying the rule to wrong scenarios, sign mistakes, and forgetting to check answers in the original problem.