## Math 9 Chapter 1 Lesson 7: Simple transformation of expressions containing square roots (continued)

## 1. Summary of theory

### 1.1. Desample of square root expression

When transforming an expression containing a square root, one can use de-sampling of the square root expression.

In a general way:

With \(A\geq 0;B\neq 0\Rightarrow \sqrt{\frac{A}{B}}=\frac{\sqrt{AB}}{|B|}\)

### 1.2. Square root axis in denominator

In a general way:

With expressions A, B where \(B>0\), we have: \(\frac{A}{\sqrt{B}}=\frac{a\sqrt{B}}{B}\)

With expressions A, B, C where \(A\geq 0, A\neq B^2\), we have \(\frac{C}{\sqrt{A}\pm B}=\frac{C (\sqrt{A}\pm B)}{AB^2}\)

With expressions A, B, C where \(A,B\geq 0;A\neq B\), we have \(\frac{C}{\sqrt{A}\pm \sqrt{B}}= \frac{C(\sqrt{A}\pm \sqrt{B})}{AB}\)

## 2. Illustrated exercise

### 2.1. Form 1: De-sampling problem of square root expression

Desample the following square-rooted expression: \(\sqrt {\frac{3}{5}} ;\,\,\sqrt {\frac{{9x}}{{7y}}} \left( { xy > 0} \right)\)

**Solution guide:**

\(\sqrt {\frac{3}{5}} = \frac{{\sqrt {3.5} }}{{\left| 5 \right|}} = \frac{{\sqrt 5 }}{5} ;\,\,\sqrt {\frac{{9x}}{{7y}}} = \frac{{\sqrt {9x.7y} }}{{\left| {7y} \right|}} = \ frac{{\sqrt {63xy} }}{{7\left| y \right|}}\)

### 2.2. Form 2: The problem of the radical axis in the form

Radial axis in the form (Assuming all expressions make sense)

\(\frac{2+\sqrt{2}}{1+\sqrt{2}}\) ; \(\frac{p-2\sqrt{p}}{\sqrt{p}-2}\)

**Solution guide:**

\(\frac{2+\sqrt{2}}{1+\sqrt{2}}=\frac{\sqrt{2}(\sqrt{2}+1)}{1+\sqrt{2}} =\sqrt{2}\)

\(\frac{p-2\sqrt{p}}{\sqrt{p}-2}=\frac{\sqrt{p}(\sqrt{p}-2)}{\sqrt{p}-2 }=\sqrt{p}\)

## 3. Practice

### 3. 1 Essay exercise

**Question 1. **Desample of a rooted expression: \(\sqrt {\frac{{10}}{{11}}} ;\,\,\sqrt {\frac{7}{{11{x^3}}}} \,\,\left( {x > 0} \right)\)

**Verse 2. **Radial axis in pattern: \(\frac{6}{{\sqrt {11} }};\,\,\frac{1}{{4\sqrt {21} }};\,\,\frac{ {3\sqrt 3 + 3}}{{6\sqrt 3 }}.\)

**Verse 3.** Radial axis in pattern: \(\frac{5}{{\sqrt 5 – 1}};\,\,\frac{3}{{\sqrt 7 – \sqrt 5 }};\,\,\frac {1}{{\sqrt m – \sqrt n }}\,\,\,\left( {m,\,n > 0} \right)\)

### 3.2 Multiple choice exercises

**Question 1. **When the radical axis of the expression \(\frac{1}{\sqrt{2}+\sqrt{3}}\) we get:

A. \(\sqrt{3}+\sqrt{2}\)

B.\(\sqrt{3}+2\)

C. \(\sqrt{3}-2\)

D. \(\sqrt{3}-\sqrt{2}\)

**Verse 2.** The reduced expression \(\frac{5+2\sqrt{6}}{5-2\sqrt{6}}\) has the value:

A. \(49+20\sqrt{6}\)

B. \(49-20\sqrt{6}\)

C. \(48-20\sqrt{6}\)

D. \(48+20\sqrt{6}\)

**Verse 3. **Solve the equation \(\sqrt {\frac{{3{\rm{x}} – 2}}{{2{\rm{x}} – 1}}} = 1\)

A. The equation has a solution of x = 0

B. The equation has a solution of x = 1

C. The equation has a solution of x = 1

D. Equation with no solution

**Verse 4.** Suppose the equation \(\frac{{\sqrt {3{\rm{x}} – 2} }}{{\sqrt {2{\rm{x}} – 1} }} = 1\)

A. The equation has a solution of x = 0

B. The equation has a solution of x = 1

C. The equation has a solution of x = -3

D. Equation with no solution

**Question 5.** Suppose the equation \(\sqrt {{{\left( {\frac{{ – 3}}{7}} \right)}^2}. {x^2}} = 3\)

A. The solution is \(x = \pm \sqrt 7 \)

B. The equation has a solution of \(x = \pm 7\)

C. The solution is \(x = \pm \frac{3}{7}\)

D. Equation with no solution

**Verse 6.** With a positive. Which of the following assertion is true?

A. \(a + \frac{1}{a} \ge 2\)

B. \(a + \frac{1}{a} \ge 4\)

C. \(a + \frac{1}{a} \le 3\)

D. \(a + \frac{1}{a} \le 3\)

**Verse 7.** Desample of the squared expression. Which of the following assertion true

A. \(\sqrt {\frac{3}{7}} = \frac{{\sqrt {21} }}{7}\)

B. \(\sqrt {\frac{{50}}{6}} = \frac{{53}}{3}\)

C. \(\sqrt {\frac{{4{\rm{a}}}}{{3b}}} = \frac{{2\sqrt {3{\rm{a}}b} }}{{ 3b}}\,\,\left( {a,b > 0} \right)\)

D. All 3 answers above are correct

**Verse 8.** With \(a = \sqrt 2 + \frac{1}{{\sqrt 2 }}\) then the value of the expression \(P = 2{{\rm{a}}^2} + 2{\rm {a}}\sqrt 2 + 1\) equals

A. 15 B. 16

C. -15 D. 16

**Verse 9. **The axis is in the sample. Which of the following assertion wrong?

A. \(\frac{3}{{\sqrt 3 + 1}} = \frac{{3\left( {\sqrt 3 – 1} \right)}}{2}\)

B. \(\frac{1}{{5 – \sqrt 5 }} = \frac{{5 + \sqrt 5 }}{{20}}\)

C. \(\frac{{\sqrt 7 – \sqrt 3 }}{{\sqrt 7 + \sqrt 3 }} = \frac{{5 + \sqrt {21} }}{2}\)

D. A, B are correct; C is wrong

**Verse 10. **Axis in the pattern: \(P = \frac{1}{{\sqrt {7 + 2\sqrt {10} } }}\)

A. \(P = \frac{{\sqrt 5 – \sqrt 2 }}{3}\)

B. \(P = \frac{{\sqrt 5 + \sqrt 2 }}{2}\)

C. \(P = \frac{{\sqrt 5 – \sqrt 3 }}{3}\)

D. \(P = \frac{{\sqrt 2 + \sqrt 3 }}{2}\)

**Verse 11. **Shorten \(M = \frac{{a – 2\sqrt a }}{{\sqrt a – 2}}\,\,\left( {a > 0} \right)\)

A. \(M = \sqrt a \)

B. \(M = a\sqrt a \)

C. \(M = -2\sqrt a \)

D. \(M = -a\sqrt a \)

## 4. Conclusion

Through this lecture on Simple transformation of expressions containing square roots, you need to complete some of the objectives given by the lesson, such as:

- Desample the square root expression.
- Square root axis in the denominator.

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