Simplifying Basic Radicals
By the end of this lesson, you’ll be able to:
- Rewrite radicals using perfect square factors.
- Break square roots into factors to simplify expressions.
- Multiply and divide radicals correctly.
- Avoid common mistakes involving radicals, exponents, and absolute values.
Key Ideas
A radical represents a root—most commonly a square root:
\[ \sqrt{a} = \text{the number whose square is } a. \]
To simplify radicals:
Break into perfect square factors
\[ \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \]
Multiply or divide under one root
\[ \sqrt{a}\sqrt{b} = \sqrt{ab} \qquad\qquad \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \]
Square roots undo squares
\[ \sqrt{x^2} = |x| \]
To fully simplify a radical:
Pull out the largest perfect square factor inside the root.
Common Problem Types
1. Simplifying Square Roots
Example:
\[
\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}
\]
2. Multiplying Radicals
Example:
\[
\sqrt{3} \cdot \sqrt{12} = \sqrt{36} = 6
\]
3. Dividing Radicals
Example:
\[
\frac{\sqrt{27}}{\sqrt{3}} = \sqrt{9} = 3
\]
4. Variables Under Radicals
Examples:
- \(\sqrt{x^4} = x^2\)
- \(\sqrt{x^3} = x\sqrt{x}\)
- \(\sqrt{x^2} = |x|\) (important when the sign matters)
5. Larger Numbers
Example:
\[
\sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}
\]
Strategies
- Use a factor tree to locate perfect squares quickly.
- Always check whether the radical can be simplified further.
- Keep variables in proper simplified form (even exponents come out cleanly).
- Multiply inside the root before simplifying if that makes the factoring easier.
Worked Examples
Example 1
Simplify:
\[
\sqrt{98}
\]
Step-by-step:
- Factor 98 → \(98 = 49 \cdot 2\)
- \(\sqrt{49} = 7\)
- Result → \(7\sqrt{2}\)
Answer: \(7\sqrt{2}\)
Example 2
Simplify:
\[
\sqrt{75} + \sqrt{12}
\]
Step-by-step:
- \(\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}\)
- \(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\)
- Combine like radicals → \(5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}\)
Answer: \(7\sqrt{3}\)
Example 3
Simplify:
\[
\sqrt{x^5}
\]
Step-by-step:
- Break the exponent into even + leftover: \(x^5 = x^4 \cdot x\)
- \(\sqrt{x^4} = x^2\)
- Leftover \(x\) stays inside → \(x^2 \sqrt{x}\)
Answer: \(x^2 \sqrt{x}\)
Example 4
Simplify:
\[
\sqrt{x^6}
\]
Step-by-step:
- Write \(x^6 = (x^3)^2\)
- \(\sqrt{(x^3)^2} = |x^3|\)
If a problem specifies that \(x>0\), then \(|x^3| = x^3\).
Otherwise leave the absolute value.
Answer (most correct): \(|x^3|\)
- Leaving radicals partially simplified by not pulling out the largest perfect square.
- Thinking \(\sqrt{a + b} = \sqrt{a} + \sqrt{b}\) (never true).
- Dropping absolute values: \(\sqrt{x^2}\) must be \(|x|\).
- Only simplifying part of the radicand (e.g., simplifying 12 but not noticing 36).
- Confusing radical and exponent rules—they interact, but they are not interchangeable.
Practice Problems
- Simplify: \(\sqrt{45}\)
- Simplify: \(\sqrt{8} \cdot \sqrt{12}\)
- Simplify: \(\frac{\sqrt{48}}{\sqrt{3}}\)
- Simplify: \(\sqrt{x^6}\)
- Combine: \(\sqrt{27} + \sqrt{3}\)
1. \(\sqrt{45}\)
- \(45 = 9 \cdot 5\)
- \(\sqrt{45} = 3\sqrt{5}\)
Answer: \(3\sqrt{5}\)
2. \(\sqrt{8} \cdot \sqrt{12}\)
- Multiply first → \(\sqrt{96}\)
- \(96 = 16 \cdot 6\) → \(\sqrt{96} = 4\sqrt{6}\)
Answer: \(4\sqrt{6}\)
3. \(\frac{\sqrt{48}}{\sqrt{3}}\)
- Combine → \(\sqrt{\frac{48}{3}} = \sqrt{16} = 4\)
Answer: \(4\)
4. \(\sqrt{x^6}\)
- \((x^3)^2\) → square root = \(|x^3|\)
Answer: \(|x^3|\)
5. \(\sqrt{27} + \sqrt{3}\)
- \(\sqrt{27} = 3\sqrt{3}\)
- Combine → \(3\sqrt{3} + \sqrt{3} = 4\sqrt{3}\)
Answer: \(4\sqrt{3}\)
Summary
- Break the radicand into perfect square factors to simplify.
- Multiply or divide inside one radical sign when helpful.
- Watch for absolute values when simplifying even powers under radicals.
- Express final results in standard simplified radical form.
- Look for perfect square factors: \(4, 9, 16, 25, 36, 49, 64,\ldots\)
- Combine like radicals only when the radicand matches.
- Use \(x^{2k} \rightarrow |x|^k\) to simplify variables under radicals.