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Step 11: Algebra 1

Radicals and rational expressions

Step 11 · Algebra 1 topics

Simplify square roots, add, multiply and rationalize radical expressions, and simplify, multiply and add rational expressions.

Simplifying radicals

To simplify a square root, take out the largest perfect square factor, using √(ab) = √a · √b.

Example: Simplifying

Simplify √72.

  1. 72 = 36 × 2, and 36 is a perfect square.
  2. √72 = √36 · √2 = 6√2.

Answer6√2

Example: With a variable

Simplify √(50x²), for x ≥ 0.

  1. 50x² = 25 · 2 · x².
  2. √25 · √(x²) · √2 = 5x√2.

Answer5x√2

Operations with radicals

Like radicals, with the same number under the root, combine like like terms: 3√2 + 5√2 = 8√2. To multiply, multiply the numbers outside and the numbers inside.

Example: Adding

Simplify √12 + √27.

  1. √12 = 2√3 and √27 = 3√3.
  2. 2√3 + 3√3 = 5√3.

Answer5√3

Example: Multiplying

Simplify (2√3)(4√6).

  1. Outside: 2 × 4 = 8. Inside: √3 · √6 = √18, which is 3√2.
  2. 8 × 3√2 = 24√2.

Answer24√2

Example: Rationalizing

Simplify 63\frac{6}{\sqrt{3}}.

  1. Multiply the top and bottom by √3: 633\frac{6\sqrt{3}}{3}.

Answer2√3

Rational expressions

A rational expression is a fraction with polynomials on the top and bottom. Simplify it by factoring both and cancelling common factors, and note the values that make the bottom 0.

Example: Simplifying

Simplify x2−9x2+5x+6\frac{x^{2} - 9}{x^{2} + 5x + 6}.

  1. Factor: (x+3)(x−3)(x+3)(x+2)\frac{(x + 3)(x - 3)}{(x + 3)(x + 2)}.
  2. Cancel x + 3, for x ≠ −3 and x ≠ −2.

Answerx−3x+2\frac{x - 3}{x + 2}

Example: Multiplying

Simplify 2xx+1⋅x2−14x2\frac{2x}{x + 1} \cdot \frac{x^{2} - 1}{4x^{2}}.

  1. Factor: 2xx+1⋅(x+1)(x−1)4x2\frac{2x}{x + 1} \cdot \frac{(x + 1)(x - 1)}{4x^{2}}.
  2. Cancel x + 1 and 2x.

Answerx−12x\frac{x - 1}{2x}

Example: Adding

Simplify 3x+2x+1\frac{3}{x} + \frac{2}{x + 1}.

  1. Use the common denominator x(x + 1): 3(x+1)+2xx(x+1)\frac{3(x + 1) + 2x}{x(x + 1)}.

Answer5x+3x(x+1)\frac{5x + 3}{x(x + 1)}

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Simplify √48.

    Show answer

    4√3

  2. Simplify √200.

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    10√2

  3. Simplify 3√5 + 2√20.

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    7√5

  4. Simplify √6 · √15.

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    3√10

  5. Simplify 105\frac{10}{\sqrt{5}}.

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    2√5

  6. Simplify x2+4xx2−16\frac{x^{2} + 4x}{x^{2} - 16}.

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    xx−4\frac{x}{x - 4}

  7. Simplify 1x+12x\frac{1}{x} + \frac{1}{2x}.

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    32x\frac{3}{2x}

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