Solutions to Exponents and Radicals Quiz

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Solutions to  http://denviant.blogspot.com/2012/04/engineering-quiz-2-exponents-and.html 

1. Solution:

3^x – 3^x-1 – 3^x-2    = 3^x - 3^x/3 - 3^x/3²

                        = 3^x (1 – 1/3 – 1/9)

                        = 3^x (5/9)

                        = 3^x (5/3²)

3^x – 3^x-1 – 3^x-2    = 5 x 3^x-2

2. Solution:

[x – √(1-x)]^1/2 = 1 - √x                square both sides

x – √(1-x) = 1 - 2√x + x              cancel x

-√(1-x) = 1 - 2√x                        square both sides

1 – x = 1 - 4√x + 4x

4√x = 5x                                   square both sides

25x² – 16x = 0

x(25x – 16) = 0

x = 0 and x = 16/25

3. Solution:

[(x²y³z^-2)^-3 (x^-3yz³)^-1/2] ÷ (xyz^-3)^5/2

        = [(x^-6y^-9z⁶) (x^3/2y^-1/2z^-3/2)] ÷ (x^-5/2y^-5/2z^15/2)

        = [x^-9/2y^-19/2z^9/2] ÷ (x^-5/2y^-5/2z^15/2)

        = 1 / x^-5/2+9/2 y^-5/2+19/2 z^15/2-9/2

        = 1 / x²y⁷z³

4. Solution:

27^x = 9^y ;   3^3x = 3^2y, thus 3x = 2y and y = 3x/2

81^y3^-x = 243

81^3x/2 3^-x = 243

(3⁴)^3x/2 3^-x = 243

3^6x 3^-x = 243

3^5x = 3⁵

5x = 5; x=1

5. Solution:

(xy^-1 / x^-2y³)⁴ ÷ (x²y^-2 / x^-3y³)³

        = (xy^-1 / x^-2y³)⁴ * (x^-3y³ / x²y^-2)³

        = x⁴y^-4 / x^-8y¹² * x^-9y⁹ / x⁶y^-6

        = x^4-9-(-8)-6 y^-4+9-12-(-6)

        = x^-3 y^-1

        = 1/x³y

6. Solution:

3^4x = (3²)^2x

3^4x = 9^2x

9^x.9^x = 5(5)

9^2x = 25

3^4x = 25

7. Solution:

{4x / [x - (x² – 4)^1/2]} * {[x + (x² – 4)^1/2] / [x + (x² – 4)^1/2]}

= [4x² + 4x (x² – 4)^1/2 ] / x² – (x² – 4)

= {4x [x + (x² – 4)^1/2]} / 4

= x[x + (x²-4)^1/2]

8. Solution:

[a^1/2 ÷ (a^1/2 - a^1/2 b^1/2)] * [(a^1/2 + a^1/2 b^1/2) / (a^1/2 + a^1/2 b^1/2)]

= (a + a√b) / (a – ab)

= [a(1 + √b)] / a(1 – b)

= (1 + b^1/2) ÷ (1 – b)

9. Solution:

3^x5^x+1 = 6^x+2;      3^x5^x5¹ = 6^x 6²

(3x5)^x 5     = 15^x 5      = 6^x 36

15^x/6^x        = (15/6)^x   = 36/5       taking the logarithms of both sides

xln (15/6)   = ln(36/5);

x = 2.1544

10. Solution:

(a^m)(aⁿ) = 100000       (1)   aⁿ/a^m = 10 (2)

Multiply (1) and (2)

(a^m)(aⁿ) * aⁿ/a^m = 100000 * 10

(aⁿ)² = 1000000

aⁿ = 1000 and a^m = 100

a^mn = (a^m)ⁿ = 100000

(100)ⁿ = 100000

n = log(100000) / log(100);  n = 2.5

aⁿ = a^2.5 = 1000; a = (1000)^1/2.5 = 15.849

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