model 3 based on positive and negative exponent Practice Questions Answers Test with Solutions & More Shortcuts

Answer:(b)

$(125)^{2/3}× (625)^{ - 1/4} = 5^x$

$5^{3×2/3}×5^{4×-1/4}=5^x$

$5^2×5^{-1}=5^x$

$5^1=5^x⇒ x=1$

Answer:(a)

$3^10 × 27^2 = 9^2 × 3^n$

$3^10 × (3^3)^2 = (3^2)^2 × 3^n$

$3^10 × 3^6 = 3^4 × 3^n$

$3^{10+6} = 3^4 × 3^n$

$3^16 = 3^{4 + n}$

4 + n = 16

n = 16 - 4 = 12

Answer:(d)

$x = (3)^{1/3} - (3)^{-1/3}$

On cubing both sides,

$x^3 = ((3)^{1/3} - (3)^{-1/3})^3$

= $((3)^{1/3})^3-((3)^{-1/3})^3-3×3^{1/3}×3^{-1/3}(3^{1/3}-3^{-1/3})$

=$3-3^{-1}-3×x=3-1/3-3x$

$x^3 + 3x = 3- 1/3={9- 1}/3$

$x^3 + 3x = 8/3$

$3x^3 + 9x = 8/3 ×3 =8$

Answer:(a)

Sum of the cubes of first n natural numbers = $({n(n + 1)}/2)^2$

Their average = ${n(n+1)^2}/4$

Required average when n = 7,

= ${7(7+ 1)^2}/4 = {7 × 8×8}/4$ = 112

Answer:(b)

$(a+b)(a^2-ab+b^2)=a^3+b^3$

$(x^{1/3}+x^{-1/3})(x^{2/3}-1+x^{-2/3})$

=$(x^{1/3}+x^{-1/3})((x^{1/3})^2-x^{1/3}.x^{-1/3}+(x^{-1/3})^2)$

=$(x^{1/3})^3+(x^{-1/3})^3$

$x+x^{-1}=x+1/x$