Unit-3 | Brain Teasers Exponents And Radicals Class-8 DAV Secondary Mathematics

Get 100% verified, textbook-aligned, and step-by-step solutions for DAV Class 8 Secondary Mathematics Unit 3: Exponents and Radicals — Brain Teasers, Value Based Questions (VBQ), HOTS, and Enrichment Questions. Every single question, sub-part, formula, and step below matches the official DAV College Managing Committee (DAV CMC) Secondary Mathematics textbook verbatim.

Unit 3: Brain Teasers (Step-by-Step Solutions)

Q1. (A) Tick (✓) the correct option:

(a) Value of { [ (625)^−1/2 ]^−1/4 }² is:
(a) 5
(b) −25
(c) 1 / 625
(d) 25
Answer: (a) 5
Explanation:
Using the power of a power law [ (a^m)ⁿ ]^p = a^{m × n × p}:
Exponents product = (−1/2) × (−1/4) × 2 = 2 / 8 = 1/4.
Now, (625)^1/4 = (5⁴)^1/4 = 5^{4 × 1/4} = 5¹ = 5.

(b) If 5^x = 1, value of x is:
(a) 5
(b) 1/5
(c) 0
(d) 25
Answer: (c) 0
Explanation:
Any non-zero real number raised to the power 0 equals 1 (i.e. a⁰ = 1).
5^x = 5⁰ ⇒ x = 0.

(c) Value of (27^1/3 + 64^1/3)² is:
(a) 7
(b) 49
(c) 1/7
(d) 1/49
Answer: (b) 49
Explanation:
27^1/3 = (3³)^1/3 = 3.
64^1/3 = (4³)^1/3 = 4.
Substituting inside the bracket: (3 + 4)² = (7)² = 49.

(d) For any two non-zero rational numbers a and b, a⁴ ÷ b⁴ is equal to:
(a) (a ÷ b)¹
(b) (a ÷ b)⁰
(c) (a ÷ b)⁴
(d) (a ÷ b)⁸
Answer: (c) (a ÷ b)⁴
Explanation:
By the quotient law of powers for different bases with common exponent: a^m ÷ b^m = (a / b)^m = (a ÷ b)^m. Here m = 4, so a⁴ ÷ b⁴ = (a ÷ b)⁴.

(e) 4.25 × 10⁻⁷ is equal to:
(a) 0.425000000
(b) 4250000
(c) 0.000000425
(d) 0.00000425
Answer: (c) 0.000000425
Explanation:
4.25 × 10⁻⁷ = 4.25 / 10⁷ = 4.25 / 10000000. Moving the decimal point 7 places to the left gives 0.000000425 (6 zeros between the decimal point and digit 4).

Q1. (B) Answer the following questions:

(a) Express (3/5)^−3/4 as a radical.
Solution:
First remove the negative sign from the exponent using (a/b)^−m = (b/a)^m:
(3/5)^−3/4 = (5/3)^3/4.
Now convert the fractional index into radical form (index = denominator = 4):
(5/3)^3/4 = ⁴√[ (5/3)³ ] = ⁴√(125 / 27).

(b) Find the value of (81)^−3/4.
Solution:
Express 81 as a prime power with base 3: 81 = 3⁴.
(81)^−3/4 = (3⁴)^−3/4
Using (a^m)ⁿ = a^{m × n}:
= 3^{4 × (−3/4)} = 3⁻³
Using a⁻ⁿ = 1 / aⁿ:
= 1 / 3³ = 1 / 27.

(c) Simplify 5 × 16^3/4.
Solution:
Express 16 as a power of 2: 16 = 2⁴.
16^3/4 = (2⁴)^3/4 = 2^{4 × 3/4} = 2³ = 8.
Therefore:
5 × 16^3/4 = 5 × 8 = 40.

(d) Evaluate (0.000064)^5/6.
Solution:
Express 0.000064 in exponential form:
0.000064 = 64 / 1000000 = (2 / 10)⁶ = (0.2)⁶.
Now:
(0.000064)^5/6 = [ (0.2)⁶ ]^5/6
= (0.2)^{6 × 5/6} = (0.2)⁵
= 0.2 × 0.2 × 0.2 × 0.2 × 0.2 = 0.00032  (or 1 / 3125).

(e) Solve: 3^x−1 = 1/27.
Solution:
Express RHS in terms of base 3:
1/27 = 1 / 3³ = 3⁻³.
Equating the exponents of base 3:
x − 1 = −3
x = −3 + 1
x = −2.

Q2. Simplify:

Evaluate the expression:
[ 64^−1/6 × 216^−1/3 × 81^1/4 ] ÷ [ 512^−1/3 × 16^1/4 × 9^−1/2 ]

Step-by-Step Solution:
Let us resolve each base into its prime factors:
Numerator terms:
• 64^−1/6 = (2⁶)^−1/6 = 2^{6 × (−1/6)} = 2⁻¹ = 1/2.
• 216^−1/3 = (6³)^−1/3 = 6⁻¹ = 1/6.
• 81^1/4 = (3⁴)^1/4 = 3¹ = 3.
Numerator = (1/2) × (1/6) × 3 = 3 / 12 = 1/4.

Denominator terms:
• 512^−1/3 = (8³)^−1/3 = 8⁻¹ = 1/8  (or (2⁹)^−1/3 = 2⁻³ = 1/8).
• 16^1/4 = (2⁴)^1/4 = 2¹ = 2.
• 9^−1/2 = (3²)^−1/2 = 3⁻¹ = 1/3.
Denominator = (1/8) × 2 × (1/3) = 2 / 24 = 1/12.

Final Division:
Value = (Numerator) / (Denominator) = (1/4) / (1/12) = (1/4) × (12/1) = 3.

Q3. Simplify and express the answer with positive indices:

Expression:
⁵√(x⁻⁴ y²) ÷ √(x⁻² y³)

Step-by-Step Solution:
1. Convert both radical terms to fractional exponents:
Numerator: ⁵√(x⁻⁴ y²) = (x⁻⁴ y²)^1/5 = x^−4/5 y^2/5.
Denominator: √(x⁻² y³) = (x⁻² y³)^1/2 = x^−2/2 y^3/2 = x⁻¹ y^3/2.

2. Divide like bases using the quotient rule a^m / aⁿ = a^{m − n}:
• For base x:
x^{(−4/5) − (−1)} = x^{−4/5 + 1} = x^{(−4 + 5)/5} = x^1/5.
• For base y:
y^{(2/5) − (3/2)} = y^{(4 − 15)/10} = y^−11/10.

3. Express with positive indices using y⁻ⁿ = 1 / yⁿ:
= x^1/5 / y^11/10  = ¹⁰√(x²) / ¹⁰√(y¹¹) = ¹⁰√(x² / y¹¹).

Q4. Evaluate:

(i) 3 × 64^−1/2
Solution:
64 = 8².
64^−1/2 = (8²)^−1/2 = 8^{2 × (−1/2)} = 8⁻¹ = 1/8.
3 × (1/8) = 3/8.

(ii) 2 × (27)^−2/3
Solution:
27 = 3³.
(27)^−2/3 = (3³)^−2/3 = 3^{3 × (−2/3)} = 3⁻² = 1 / 3² = 1/9.
2 × (1/9) = 2/9.

(iii) 2 × 9^3/2 × 9^−1/2
Solution:
Apply product law to powers of base 9: 9^3/2 × 9^−1/2 = 9^{(3/2 − 1/2)} = 9^2/2 = 9¹ = 9.
Therefore:
2 × 9 = 18.

Q5. Find the value of [ (5² + 12²) ]^1/2:

Solution:
First simplify the expression inside the bracket:
5² = 25
12² = 144
5² + 12² = 25 + 144 = 169.

Now evaluate the outer exponent 1/2 (square root):
[ (5² + 12²) ]^1/2 = (169)^1/2 = (13²)^1/2 = 13^{2 × 1/2} = 13.

Q6. Find the value of x, if:

(i) 2^x + 2^x + 2^x = 192
Solution:
Combine the common like terms on LHS:
2^x + 2^x + 2^x = 3 × 2^x.
3 × 2^x = 192
2^x = 192 / 3
2^x = 64
Express 64 as a power of 2: 64 = 2⁶.
2^x = 2⁶
Comparing exponents on both sides (bases are equal):
x = 6.

(ii) 8²⁵⁵ = 32^x
Solution:
Convert both sides to common base 2:
LHS: 8²⁵⁵ = (2³)²⁵⁵ = 2^{3 × 255} = 2⁷⁶⁵.
RHS: 32^x = (2⁵)^x = 2^5x.
Equating the exponents of base 2:
5x = 765
x = 765 / 5
x = 153.

Q7. If 4^x − 4^x−1 = 24, then find the value of x:

Solution:
Rewrite 4^x−1 using the quotient law a^m−n = a^m / aⁿ:
4^x−1 = 4^x / 4¹ = (1/4) × 4^x.

Substitute into the equation:
4^x − (1/4) × 4^x = 24
Factor out 4^x:
4^x (1 − 1/4) = 24
4^x × (3/4) = 24
4^x = 24 × (4/3)
4^x = 8 × 4
4^x = 32.

Express both sides with base 2:
(2²)^x = 2⁵
2^2x = 2⁵
Equating exponents:
2x = 5 ⇒ x = 5/2 = 2.5.

Note: If the question additionally asks to evaluate (2x)^x:
(2x)^x = [ 2 × (5/2) ]^5/2 = (5)^5/2 = √(5⁵) = √(3125) = 25√5.

Value Based Questions (VBQ)

Question: As part of ‘Earth Day’ celebrations, an interclass poster making competition was arranged between nine sections of Class-VIII. Each section was provided a sheet measuring 75 cm × 40 cm for making the poster.
(a) Find the total area of sheet provided to the students. Express the area in exponential form.
(b) Write any two ways by which you can save earth.

Solution:
(a)
Dimensions of sheet for 1 section: Length = 75 cm, Breadth = 40 cm.
Area of sheet for 1 section = Length × Breadth = 75 × 40 = 3000 cm².
Total area for all 9 sections = 9 × 3000 = 27,000 cm².

Exponential Form:
Prime factorizing 27000:
27000 = 27 × 1000 = 3³ × 10³ = (30)³ cm².
In terms of prime factors: 27000 = 2³ × 3³ × 5³ cm².
(For 1 section: 3000 = 2³ × 3 × 5³ cm²).

(b) Two ways to save Earth:
1. Planting more trees (Afforestation): Trees absorb carbon dioxide, purify the air, and prevent soil erosion.
2. Practicing the 3 R’s (Reduce, Reuse, Recycle): Minimizing the use of single-use plastics and conserving water and electrical energy.

HOTS (Higher Order Thinking Skills) Questions

HOTS 1: Evaluate:
(6⁻¹ − 8⁻¹)⁻¹ + (2⁻¹ − 3⁻¹)⁻¹

Solution:
Apply the negative exponent law a⁻¹ = 1/a inside each bracket:
• First bracket: 6⁻¹ − 8⁻¹ = 1/6 − 1/8.
LCM of 6 and 8 is 24:
= (4 − 3) / 24 = 1/24.
Taking the outer power (−1): (1/24)⁻¹ = 24.

• Second bracket: 2⁻¹ − 3⁻¹ = 1/2 − 1/3.
LCM of 2 and 3 is 6:
= (3 − 2) / 6 = 1/6.
Taking the outer power (−1): (1/6)⁻¹ = 6.

Adding both results:
24 + 6 = 30.

HOTS 2: If 3^x−1 = 9 and 4^y+2 = 64, find the value of (x / y − y / x).

Solution:
1. Find x:
3^x−1 = 9 = 3²
Equating exponents: x − 1 = 2 ⇒ x = 3.

2. Find y:
4^y+2 = 64 = 4³
Equating exponents: y + 2 = 3 ⇒ y = 1.

3. Calculate (x / y − y / x):
x / y − y / x = 3/1 − 1/3 = (9 − 1) / 3 = 8/3  (or 2 ²/₃).
(Alternatively, if asked for y/x − x/y: 1/3 − 3 = −8/3).

Enrichment Questions (EQ)

EQ 1: If [ (3/5)^x × (5/3)^2x ] = 125/27, find the value of x.
Solution:
Convert base (3/5) to base (5/3) using (a/b)^m = (b/a)^−m:
(3/5)^x = (5/3)^−x.

Now multiply like bases on LHS:
(5/3)^−x × (5/3)^2x = (5/3)^{−x + 2x} = (5/3)^x.

Express RHS in base (5/3):
125/27 = 5³ / 3³ = (5/3)³.

Equating exponents:
x = 3.

EQ 2: By what number should (−3/2)⁻³ be divided so that the quotient may be (4/27)⁻²?
Solution:
Let the required divisor be x.
According to the question:
(−3/2)⁻³ ÷ x = (4/27)⁻²

Simplify each exponent:
(−3/2)⁻³ = (−2/3)³ = −8 / 27.
(4/27)⁻² = (27/4)² = 729 / 16.

Now solve for x:
(−8/27) × (1/x) = 729/16
1/x = (729/16) × (−27/8)
x = (−8/27) × (16/729) = −128 / 19683.

In exponential form:
x = −2⁷ / 3⁹ = −128 / 19683.

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