Square Roots

Worked example

1. 1

   $7\times 7=49$ — Start with 7 and square it: 7 times itself is 49.

2. 2

   $\sqrt{49}=7$ — The square root goes backwards: 7 is the number that makes 49 when you multiply it by itself.

Worked example

1. 1

   $9^2=81$ — Find the biggest square number below 90. It is 81, which is 9 squared.

2. 2

   $10^2=100$ — Find the square just above 90. 100 is 10 squared.

3. 3

   $81<90<100$ — 90 sits between the two squares. A bigger number has a bigger root, so the roots keep the same order.

4. 4

   $9<\sqrt{90}<10$ — Take the root of each part. So √90 is between 9 and 10.

5. 5

   $\sqrt{90}=9.486\dots$ — Check with the root key: 9.486… is between 9 and 10, as expected.

Worked example

1. 1

   $40^2=1600$ — 1764 is more than 1600, so its root is more than 40.

2. 2

   $50^2=2500$ — 1764 is less than 2500, so its root is less than 50. The answer is in the 40s.

3. 3

   $\text{ends in }4:2\text{ or }8$ — Look at the last digit. Only 2 × 2 = 4 and 8 × 8 = 64 end in 4, so the root ends in 2 or 8.

4. 4

   $45^2=2025$ — Choose between 42 and 48 by squaring the middle, 45. 2025 is bigger than 1764, so the root is below 45: it is 42.

5. 5

   $42\times 42=1764$ — Check by squaring back. It gives 1764, so $\sqrt{1764}=42$.

Worked example

1. 1

   $8^2=64$ — 70.56 is more than 64, so its root is more than 8.

2. 2

   $9^2=81$ — 70.56 is less than 81, so its root is less than 9. The answer is 8 point something.

3. 3

   $\sqrt{70.56}=8.4$ — Press the root key, then type 70.56. 8.4 is between 8 and 9, so it fits the estimate.

4. 4

   $8.4\times 8.4=70.56$ — Check: square it back. It gives 70.56, the number you started with, so 8.4 is right.

Worked example

1. 1

   $\sqrt{\frac{16}{49}}=\frac{\sqrt{16}}{\sqrt{49}}$ — Split into the top and the bottom. Both are square numbers.

2. 2

   $\sqrt{16}=4$ — 4 × 4 = 16, so the root of the top is 4.

3. 3

   $\sqrt{49}=7$ — 7 × 7 = 49, so the root of the bottom is 7.

4. 4

   $\sqrt{\frac{16}{49}}=\frac{4}{7}$ — Put them back together. Check: $\frac{4}{7}\times \frac{4}{7}=\frac{16}{49}$.

Worked example

1. 1

   $1.69=\frac{169}{100}$ — 1.69 has 2 decimal places, so it is 169 hundredths. 100 is a square: 10 × 10.

2. 2

   $\sqrt{\frac{169}{100}}=\frac{13}{10}$ — Root of the top is 13, because 13 × 13 = 169. Root of the bottom is 10.

3. 3

   $\frac{13}{10}=1.3$ — Change back to a decimal. Check: 1.3 × 1.3 = 1.69.

Worked example

1. 1

   $\sqrt{2}\times \sqrt{32}=\sqrt{2\times 32}$ — Neither 2 nor 32 is a square, so join them under one root sign.

2. 2

   $=\sqrt{64}$ — 2 × 32 = 64, which is a square number.

3. 3

   $=8$ — 8 × 8 = 64. Check on a calculator: 1.414… × 5.656… = 8.

Worked example

1. 1

   $\sqrt{16\times 81}=\sqrt{16}\times \sqrt{81}$ — Both 16 and 81 are squares, so split it into two roots. This avoids working out 16 × 81.

2. 2

   $=4\times 9$ — 4 × 4 = 16 and 9 × 9 = 81.

3. 3

   $=36$ — Check: 36 × 36 = 1296, and 16 × 81 = 1296 too.

Worked example

1. 1

   $3x^2=48$ — The equation. The $x^2$ has a 3 in front, so get $x^2$ alone first.

2. 2

   $x^2=16$ — Divide both sides by 3, because 48 ÷ 3 = 16.

3. 3

   $x=4$ — Take the square root of both sides: 4 × 4 = 16.

4. 4

   $(-4{)}^2=(-4)\times (-4)=16$ — But −4 works too, because minus times minus is plus.

5. 5

   $x=4\text{or}x=-4$ — So there are two answers. Check: 3 × 16 = 48.

Worked example

1. 1

   $l^2=56.25$ — Area of a square = side × side. Put in the area.

2. 2

   $l=7.5\text{or}l=-7.5$ — Take the square root of both sides. Both 7.5 and −7.5 square to 56.25.

3. 3

   $l=7.5\text{ cm}$ — A length cannot be negative, so reject −7.5. The area is in cm², so the side is in cm.

4. 4

   $7.5\times 7.5=56.25$ — Check by squaring back. It matches the area.