Finding the Largest Perfect Square Factor

Simplify $\sqrt{72}$

List perfect squares less than 72: $1, 4, 9, 16, 25, 36, 49, 64$ = Perfect squares to check

Find which divide 72 evenly: $72 \div 4 = 18$ \checkmark, $72 \div 9 = 8$ \checkmark, $72 \div 36 = 2$ \checkmark = 4, 9, and 36 are factors

Choose the largest: 36: $\sqrt{72} = \sqrt{36 \times 2}$ = Factor out 36

Apply the product rule: $\sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}$ = Separate the roots

Simplify the perfect square: $\sqrt{36} = 6$ = $6\sqrt{2}$

Answer: $\sqrt{72} = 6\sqrt{2}$

Using Prime Factorization

Simplify $\sqrt{180}$

Find the prime factorization: $180 = 2^2 \times 3^2 \times 5$ = Prime factors

Group pairs of factors: $180 = (2^2) \times (3^2) \times 5$ = Pairs are perfect squares

Take square root of pairs: $\sqrt{2^2} = 2$, $\sqrt{3^2} = 3$ = Each pair gives one factor outside

Multiply factors outside: $2 \times 3 = 6$ = 6 comes out of the radical

Write final answer: $\sqrt{180} = 6\sqrt{5}$ = 5 stays under the radical

Answer: $\sqrt{180} = 6\sqrt{5}$

Already Simplified

Simplify $\sqrt{15}$

Find the prime factorization: $15 = 3 \times 5$ = Two different primes

Check for pairs: No factor appears twice = No perfect square factors

Conclusion: 15 has no perfect square factors other than 1 = Already in simplest form

Answer: $\sqrt{15}$ is already in simplest form

Larger Numbers

Simplify $\sqrt{288}$

Find the prime factorization: $288 = 2^5 \times 3^2 = 2^4 \times 2 \times 3^2$ = Regroup to show pairs

Identify perfect square factors: $2^4 = 16$ and $3^2 = 9$, so $16 \times 9 = 144$ = $288 = 144 \times 2$

Apply the square root: $\sqrt{288} = \sqrt{144 \times 2} = \sqrt{144} \times \sqrt{2}$ = Separate perfect square

Simplify: $\sqrt{144} = 12$ = $12\sqrt{2}$

Answer: $\sqrt{288} = 12\sqrt{2}$

Mistake: Stopping too early: writing $\sqrt{72} = 2\sqrt{18}$ instead of $6\sqrt{2}$

Why: 18 still has a perfect square factor (9). Always check if the remaining radicand can be simplified further.

Correct: Continue simplifying: $2\sqrt{18} = 2\sqrt{9 \times 2} = 2 \times 3\sqrt{2} = 6\sqrt{2}$

Mistake: Writing $\sqrt{50} = 25\sqrt{2}$ instead of $5\sqrt{2}$

Why: We take the SQUARE ROOT of 25, not 25 itself. $\sqrt{25} = 5$, not 25.

Correct: $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$

Mistake: Thinking $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$

Why: Square roots do NOT distribute over addition! Only over multiplication.

Correct: $\sqrt{9 + 16} = \sqrt{25} = 5$, but $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$. These are not equal!

Diagonal of a Square

Finding the diagonal of a square uses the Pythagorean theorem and requires simplifying square roots.

A square has side length 6 cm. Its diagonal is $\sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2}$ cm.

Screen Sizes

TV and monitor sizes are measured diagonally, which involves square roots.

A monitor is 40 cm wide and 30 cm tall. The diagonal is $\sqrt{40^2 + 30^2} = \sqrt{2500} = 50$ cm.

Q: How do I know when a square root is fully simplified?

A: Check the number under the radical. If its only perfect square factor is 1 (no repeated prime factors), it is fully simplified. For example, $\sqrt{30}$ is simplified because $30 = 2 \times 3 \times 5$ has no repeated primes.

Q: Why do we use the largest perfect square factor?

A: Using the largest perfect square factor gets you to the answer in one step. You can use smaller factors, but you will need to simplify multiple times. Both methods give the same final answer.

Q: Can all square roots be simplified?

A: No. If the number under the radical has no perfect square factors other than 1 (like 2, 3, 5, 6, 7, 10, etc.), the square root is already in simplest form.