Math Buddyv1.0

Simplify radical expressions with steps

Type or scan a radical expression. MathBuddy pulls out perfect squares, combines like radicals and rationalizes denominators.

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What this page covers

  • Square roots and cube roots of whole numbers, written in simplest radical form.
  • Adding and subtracting like radicals once each one is simplified.
  • Rationalizing a denominator, including a two-term denominator such as 3−53 - \sqrt{5}.
  • Not covered here: a decimal value as the main answer (use Arithmetic calculator), and roots of variable expressions beyond simple cases, which you can still type and try.

How to enter the problem

  • Type sqrt(50) or use the root key on the math keyboard; for a cube root, use the nth-root key.
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1

Problem

Simplify 180\displaystyle \sqrt{180}.

Answer

Verified

The answer is

656\sqrt{5}

Explanation

  1. Find the largest perfect square factor
    180=36⋅5180 = 36 \cdot 5, and 36 is a perfect square.
  2. Split the root
    180=36⋅5\sqrt{180} = \sqrt{36} \cdot \sqrt{5}
  3. Take the square root of 36
    656\sqrt{5}. The 5 has no square factor left, so the form is simplest.

Example 2

Problem

Simplify 320−45+80\displaystyle 3\sqrt{20} - \sqrt{45} + \sqrt{80}.

Answer

Verified

The answer is

757\sqrt{5}

Explanation

  1. Simplify each radical
    20=25\sqrt{20} = 2\sqrt{5}, 45=35\sqrt{45} = 3\sqrt{5} and 80=45\sqrt{80} = 4\sqrt{5}.
  2. Rewrite the expression
    3⋅25−35+45=65−35+453 \cdot 2\sqrt{5} - 3\sqrt{5} + 4\sqrt{5} = 6\sqrt{5} - 3\sqrt{5} + 4\sqrt{5}
  3. Combine like radicals
    Every term is a multiple of 5\sqrt{5}, and 6−3+4=76 - 3 + 4 = 7.

Example 3

Problem

Rationalize and simplify 43−5\displaystyle \dfrac{4}{3 - \sqrt{5}}.

Answer

Verified

The answer is

3+53 + \sqrt{5}

Explanation

  1. Multiply by the conjugate
    Multiply the top and the bottom by 3+53 + \sqrt{5}.
  2. Expand the denominator
    (3−5)(3+5)=9−5=4(3 - \sqrt{5})(3 + \sqrt{5}) = 9 - 5 = 4
  3. Simplify
    4(3+5)4=3+5\frac{4(3 + \sqrt{5})}{4} = 3 + \sqrt{5}

Common mistakes

  • Splitting a root over a sum: 9+16=5\sqrt{9 + 16} = 5, not 9+16=7\sqrt{9} + \sqrt{16} = 7.
  • Pulling out a square that is not the largest, such as 180=320\sqrt{180} = 3\sqrt{20}, and stopping there.
  • Adding unlike radicals: 2+3\sqrt{2} + \sqrt{3} does not equal 5\sqrt{5}.
  • Multiplying only the denominator by the conjugate. The numerator has to be multiplied too.

Checks, assumptions and limits

  • Compare decimals. 50+18−8≈8.485\sqrt{50} + \sqrt{18} - \sqrt{8} \approx 8.485, and 62≈8.4856\sqrt{2} \approx 8.485 as well.
  • Simplest form means no perfect square factor under a square root and no root left in a denominator.
  • Square roots here are of non-negative numbers; the square root of a negative number is not real.
  • MathBuddy can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

The number under the root has no perfect square factor (no perfect cube for a cube root), no fraction sits under the root, and no root is left in a denominator.

Yes. The steps multiply by the root or by the conjugate and show the new denominator before simplifying.

Yes. Ask a follow-up in the same thread for a decimal approximation. The exact radical form stays the main answer.

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Last updated: · MathBuddy