Math 9 Chapter 1 Lesson 1: Square Root

1. Summary of theory

1.1. Arithmetic square root

– The square root of a non-negative number a is the number x such that \(x^2=a\)

– For a positive number a, the number \(\sqrt{a}\) is called the arithmetic square root of a.

– The number 0 is also called the arithmetic square root of 0 .

1.2. Compare two arithmetic square roots

Theorem: For two non-negative numbers a and b, we have: \(a < b \Leftrightarrow \sqrt a < \sqrt b \)

2. Illustrated exercise

2.1. Form 1: Find the arithmetic square root of a number

Find the arithmetic square root of the following numbers: 121; 144; 361; 400

Solution guide

\(\sqrt{121}=11\) because \(11 > 0\) and \(11^2=121\)

Similarly, we have: \(\sqrt{144}=12; \sqrt{361}=19; \sqrt{400}=20\)

2.2. Form 2: Compare two arithmetic square roots

Compare:

2 and \(\sqrt{3}\); 7 and \(\sqrt{51}\)

Solution guide

We have \(2=\sqrt{4}\) and \(4>3\) so \(\sqrt{4}>\sqrt{3}\) ie \(2> \sqrt{3}\)

Similarly, we have \(7=\sqrt{49}\) and \(51>49\) so \(\sqrt{49}<\sqrt{51}\) ie \(7<\sqrt{51 }\)

2.3 Form 3: Find the square root of a number

Solve the following equations: \(x^2=196\) ; \(x^2=1.69\)

Solution guide

\(x^2=196\Rightarrow x=\pm \sqrt{196}=\pm 14\)

\(x^2=1.69\Rightarrow x=\pm \sqrt{1,69}=\pm 1,3\)

2.4. Type 4: Advanced exercises involving square roots

Question 1: Find the number x that is not negative: \(2\sqrt{x}=14\) ; \(\sqrt{3x}<2\)

Solution guide

\(2\sqrt{x}=14\Leftrightarrow \sqrt{x}=7\Leftrightarrow x=49\)

\(\sqrt{3x}<2\Leftrightarrow 3x<4\Leftrightarrow x<\frac{4}{3}\) where \(x\geq 0\) should be \(0\leq x\leq \frac{4 }{3}\)

Verse 2: Calculate the side of a square, given that its area is equal to the area of ​​a rectangle whose length is 18 cm and breadth is 2 cm.

Solution guide

The area of ​​the rectangle is \(18.2=36 (cm^2)\)

Let the side length of the square be a \((a>0)\), by title, \(a^2=36\Leftrightarrow a=6(cm)\) because \(a>0\)

3. Practice

3.1. Essay exercises

Question 1: Find the arithmetic square root of the following numbers: 169; 196; 441; 0.25.

Verse 2: Compare: 3 and \(\sqrt 8 ;\) 9 and \(\sqrt {87}\) .

Question 3: Solve the following equation: \({x^2} = 144;\,\,{x^2} = 2.25\)

Question 4: Find a known non-negative number x: \(3\sqrt x = 15;{\rm{ }}\sqrt {4x} < 3\)

Question 5: Quiz. Find the side of a square, knowing that its area is equal to the area of ​​a rhombus whose diagonals are 8cm and 16cm.

3.2. Multiple choice exercises

Question 1: Find the arithmetic square root of numbers: 0.01; 0.49; 0.0081; 0.000064. Which of the following assertion wrong?

A. \(\sqrt {0.01} = 0.1.\)

B. \(\sqrt {0.49} = 0.7.\)

C. \(\sqrt {0.0081} = 0.009.\)

D. \(\sqrt {0.000064} = 0.008.\)

Verse 2: Which of the following assertion true?

A. The square root of 121 is 11.

B. The square root of 144 is 12.

C. \(\sqrt {169} = \pm 13\)

D. The square root of 225 is 15 and -15.

Question 3: Find x, know (round up to second decimal place)

A. x_first = 2.65 and x₂ = -2.65

B. x_first = 2.83 and x₂ = -2.82

C. x_first = 3.14 and x₂ = -3.14

D. A, B are both wrong.

Question 4: The equation 3 x 2= 483 x 2 = 48 has a solution of:

A. 4

B. \(-4\)

C. 8

D. \( \pm 4\)

Question 5: A square piece of land has the same area as a rectangular piece of land with a length of 25m and a width of 4m. So, the side of the square is equal to?

A. 10m

B. 20m

C. 5m

D. 15m

4. Conclusion

Through this lesson, you should know the following:

• Define arithmetic square root.
• Compare arithmetic square roots.