Radicals &
Absolute Values
Master the rules of square roots, cube roots, and absolute value expressions to unlock essential SAT Math points.
SAT Math
What Is a Radical?
A radical undoes an exponent. The expression \( \sqrt[n]{a} \) asks: "What number, raised to the \( n \)th power, gives \( a \)?"
\( \sqrt{25} = 5 \)
Because \( 5^2 = 25 \)
\( \sqrt[3]{27} = 3 \)
Because \( 3^3 = 27 \)
\( \sqrt{x^2} = |x| \)
The link to absolute value
What Is Absolute Value?
Absolute value measures the distance from a number to zero on the number line. Distance is always non-negative.
Why Together?
Radicals and absolute values are linked by one key identity that the SAT loves to test:
1
If \( x = 5 \): \( \sqrt{5^2} = \sqrt{25} = 5 = |5| \) ✓
2
If \( x = -5 \): \( \sqrt{(-5)^2} = \sqrt{25} = 5 = |-5| \) ✓
3
Without \( |\ | \): writing \( \sqrt{x^2} = x \) would give \( -5 \) for the second case, which is wrong.
Essential Radical Rules
1
Product rule: \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \) when \( a, b \geq 0 \)
2
Quotient rule: \( \sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}} \) when \( a \geq 0,\ b > 0 \)
3
Rational exponent: \( \sqrt[n]{a} = a^{1/n} \) and \( \sqrt[n]{a^m} = a^{m/n} \)
4
Simplifying: \( \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \)
Essential Absolute Value Rules
1
Non-negativity: \( |x| \geq 0 \) for all real \( x \)
2
Products: \( |ab| = |a| \cdot |b| \)
3
Quotients: \( \left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|} \) when \( b \neq 0 \)
4
Triangle inequality: \( |a + b| \leq |a| + |b| \)
Radicals vs. Absolute Values
| Radicals | Absolute Values | |
|---|---|---|
| Symbol | \( \sqrt{\ } \) or \( \sqrt[n]{\ } \) | \( |\ | \) |
| Meaning | Inverse of exponentiation | Distance from zero |
| Output sign | Non-negative (even root) | Always non-negative |
| Splits over | Multiplication, division | Multiplication, division |
| Does NOT split over | Addition, subtraction | Addition, subtraction |
| Connection | \( \sqrt{x^2} = |x| \) | |
Graph of \( y = |x| \)
Graph of \( y = \sqrt{x} \)
Step-by-Step Strategy
1
Identify the type: Is it a radical expression, an absolute value equation, or both?
2
Isolate: Get the radical or absolute value alone on one side of the equation.
3
Remove the operation: Square both sides for \( \sqrt{\ } \), or split into two cases for \( |\ | \).
4
Solve the resulting equation(s).
5
Check for extraneous solutions: Plug answers back into the original equation. Squaring can introduce false solutions.
Example 1: Simplifying Radicals
Simplify: \( \sqrt{72} \)
1
Factor out the largest perfect square: \( 72 = 36 \times 2 \)
2
Apply the product rule: \( \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} \)
3
Evaluate: \( 6\sqrt{2} \)
Example 2: Absolute Value Equation
Solve: \( |2x - 5| = 9 \)
1
Split into two cases:
Case A: \( 2x - 5 = 9 \)
Case B: \( 2x - 5 = -9 \)
Case A: \( 2x - 5 = 9 \)
Case B: \( 2x - 5 = -9 \)
2
Solve Case A: \( 2x = 14 \implies x = 7 \)
3
Solve Case B: \( 2x = -4 \implies x = -2 \)
Example 3: Radical Equation
Solve: \( \sqrt{3x + 1} = x - 1 \)
1
Square both sides: \( 3x + 1 = (x-1)^2 = x^2 - 2x + 1 \)
2
Rearrange: \( 0 = x^2 - 5x \implies x(x-5) = 0 \)
3
Candidates: \( x = 0 \) or \( x = 5 \)
Common Mistakes
✕ Splitting over addition
Writing \( \sqrt{a+b} = \sqrt{a} + \sqrt{b} \). This is false. The product rule does not apply to sums.
✕ Forgetting two cases
Solving \( |x| = 5 \) as only \( x = 5 \). You must also consider \( x = -5 \).
✕ Skipping the check
Squaring both sides can create extraneous solutions. Always plug answers back in.
Quick Reference Table
| Situation | Method | Watch Out |
|---|---|---|
| Simplify \( \sqrt{n} \) | Factor out perfect squares | Use largest factor |
| \( \sqrt{x^2} \) | Equals \( |x| \), not \( x \) | Sign matters |
| \( |f(x)| = k \) | Split: \( f(x) = k \) or \( f(x) = -k \) | No solution if \( k < 0 \) |
| \( \sqrt{f(x)} = g(x) \) | Square both sides, solve | Check for extraneous |
| Fraction like \( \frac{\sqrt{a}}{\sqrt{b}} \) | Rewrite as \( \sqrt{\frac{a}{b}} \) or multiply by \( \frac{\sqrt{b}}{\sqrt{b}} \) | Final answer has no radical in denominator |
Question 1 of 5
What is the simplified form of \( \sqrt{200} \)?
Question 2 of 5
How many solutions does \( |4x + 1| = -5 \) have?
Question 3 of 5
If \( \sqrt{x+7} = x - 5 \), what is the solution set?
Question 4 of 5
Which expression is equivalent to \( \sqrt{x^2} \) for all real numbers \( x \)?
Question 5 of 5
What is the sum of all values of \( x \) that satisfy \( |3x - 6| = 12 \)?