Simplifying square roots | Exponents, radicals, and scientific notation | Pre-Algebra | Khan Academy

Main ideas / lessons

• To simplify expressions involving square roots (e.g., (k\sqrt{n})), look for perfect-square factors inside the radicand (n).
• This is done by using prime factorization and extracting repeated prime factors.
• Use exponent (radical) properties to rewrite:
  • (\sqrt{a\cdot a} = \sqrt{a^2} = a)
  • More generally, (\sqrt{p^2\cdot q} = p\sqrt{q})

Method (step-by-step instructions)

Simplifying (5\sqrt{117})

1. Prime factorize 117

   • Check if factors are perfect squares (i.e., look for repeated primes).
   • Determine divisibility:
     • Not divisible by 2 (odd).
     • Divisible by 3 (digit sum: (1+1+7=9), and (9) is divisible by 3).
   • Divide to confirm:
     • (117 = 3 \times 39)
   • Factor further:
     • (39 = 3 \times 13)
   • So:
     • (117 = 3 \times 3 \times 13 = 3^2 \cdot 13)

2. Rewrite the square root using exponent/radical properties

   • [ \sqrt{3^2\cdot 13} = \sqrt{3^2}\cdot \sqrt{13} = 3\sqrt{13} ]

3. Multiply outside the radical

   • [ 5\sqrt{117} = 5\cdot 3\sqrt{13} = 15\sqrt{13} ]

Simplifying (3\sqrt{26})

1. Prime factorize 26

   • Since 26 is even:
     • (26 = 2 \times 13)

2. Check for perfect squares inside the radicand

   • Both 2 and 13 are prime and appear only once → no repeated prime factors → no perfect-square factor can be extracted.

3. Conclude simplest form

   • Leave it as:
     • (3\sqrt{26})

Key concept takeaway

• You simplify (\sqrt{n}) by extracting any perfect-square factors from (n); otherwise, the radical form is already as simplified as possible.

Speakers / sources featured

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• Source/context: Khan Academy (implied by the video title).