Math 9 Chapter 1 Lesson 3: Relationship between multiplication and squares

1. Summary of theory

1.1. Theorem

– Theorem: For two non-negative numbers a and b, we have: \(\sqrt{a}.\sqrt{b}=\sqrt{ab}\)

– Note: The theorem above can be extended to many non-negative numbers.

1.2. Apply

a. Rule of square one product

To square a product of non-negative numbers, we can square each factor and then multiply the results.

b. Rule for multiplying square roots

To multiply the square roots of non-negative numbers, we can multiply the numbers under the root sign together and then square the result.

Note: In general, for two non-negative expressions A and B, we have: \(\sqrt{A}.\sqrt{B}=\sqrt{AB}\)

2. Illustrated exercise

2.1. Form 1: Problem using square of a product

Question 1: Applying the rule of squaring a product, calculate:

\(\sqrt{0.09.64}\) ; \(\sqrt{2^4.(-7)^2}\)

Solution guide

We have \(\sqrt{0.09.64}=\sqrt{0.09}.\sqrt{64}=0,3.8=2,4\)

\(\sqrt{2^4.(-7)^2}=\sqrt{2^4}.\sqrt{(-7)^2}=4.7=28\)

Verse 2: Reduce the expression \(\sqrt{a^4(3-a)^2}\) to \(a\geq 3\)

Solution guide

\(\sqrt{a^4(3-a)^2}=a^2.|3-a|=a^2(a-3)\) because \(a\geq 3\)

Question 3: Calculus 12.30.40.

Solution guide

\(\sqrt{12.30.40}=\sqrt{12.3.2.2.100}=6.2.10=120\)

2.2. Type 2: Problem using the rule for multiplying square roots

Question 1: Applying the multiplication rule, calculate: \(\sqrt{7}.\sqrt{63}\) ; \(\sqrt{0,4}.\sqrt{6,4}\)

Solution guide

We have: \(\sqrt{7}.\sqrt{63}=\sqrt{7.63}=\sqrt{7.7.3.3}=7.3=21\)

\(\sqrt{0.4}.\sqrt{6,4}=\sqrt{0,4.6,4}=\sqrt{0.04.64}=\sqrt{0.04}.\sqrt{64}= 0,2.8=1.6\)

Verse 2: Calculate the value of \((2-\sqrt{3})(2+\sqrt{3})\)

Solution guide

\((2-\sqrt{3})(2+\sqrt{3})=2^2-(\sqrt{3})^2=4-3=1\)
or: \((2-\sqrt{3})(2+\sqrt{3})=2.2+2\sqrt{3}-2\sqrt{3}-\sqrt{3}.\sqrt{3} =1\)

3. Practice

3.1 Essay exercises

Question 1: Applying the square one product rule, calculate: \(\sqrt {0.04.81} \) ; \(\sqrt {{3^4}. {{( – 5)}^2}} \)

Verse 2: Applying the multiplication rule, calculate: \(\sqrt 5 .\sqrt {20} \) ; \(\sqrt {0.4} .\sqrt {3,6} \)

Question 3: Compact expression \(\sqrt {{x^8}{{(x – 2)}^2}} \) with \(x \le 2\)

Question 4: Calculus 20.24.60

Question 5: Calculate the value of \((3 – \sqrt 5 )(3 + \sqrt 5 )\)

3.2 Multiple choice exercises

Question 1. Which of the following assertion is true?

A. \(\sqrt 5 .\sqrt {80} = 20\)

B. \(\sqrt {90,6.4} = 24\)

C. \(\sqrt {21,{8^2} – 18,{2^2}} = 12\)

D. A, B, C are all correct

Verse 2. Calculate \(M = \sqrt {117,{5^2} – 26,{5^2} – 1440} \)

A. 108

B. 110

C. 120

D. 135

Verse 3. Calculate \(N = \sqrt {146,{5^2} – 109,{5^2} – 27.256} \)

A. 96

B. 108

C. 128

D. \(16\sqrt {10} \)

Verse 4. Compare \(\sqrt{25+9}\) and \(\sqrt{25}+\sqrt{9}\)

A. \(\sqrt{25+9}<\sqrt{25}+\sqrt{9}\)

B. \(\sqrt{25+9}=\sqrt{25}+\sqrt{9}\)

C. \(\sqrt{25+9}>\sqrt{25}+\sqrt{9}\)

D. \(\sqrt{25+9}.(\sqrt{25}+\sqrt{9})=1\)

Question 5. The value of the expression \(\sqrt{4(1+6x+9x^2)^2}\) at \(x=-\sqrt{2}\) is

A.\(19+6\sqrt{2}\)

B.\(19-6\sqrt{2}\)

C.\(38-12\sqrt{2}\)

D. \(38+12\sqrt{2}\)

4. Conclusion

Through this lesson, you should meet the following requirements:

Know the rules of squaring a product, the rules of multiplying square roots.
Do related maths.

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