Simplify Exponents and Radicals Questions

Review and use the the rules for radicals and exponents to simplify exponents and radical expressions; questions with detailed solutions (lower part of page) and explanations are presented. Try not to use the calculator to simplify numerical expressions except to check your answers. More questions on how to simplify radical expressions are included n this website.

Simplify Numerical Expressions with Exponents

Question 1

1. \( 3^2 \)
2.   - 3 ⁴
3.   (- 3) ⁴
4.   12 ⁰
5.   0 ²⁰
6.   0 ^{- 4}
7.   (-1) ⁷
8.   (-1) ⁴
9.   (-1) ^{- 3}
10.   (- 2 / 3) ^-2
11.   (- 2) ^-2 / 4 ^-2
12.   - 3 ^-3 + (- 2) ^-2
13.   x ⁴ for x = - 2
14.   - x ² for x = - 3
15.   - x ⁵ for x = - 2
16.   x ⁴ for x = - 2 / 3
17.   - x ^-2 for x = - 1 / 2
18.   - x ⁵ for x = - 1 / 3

Simplify Numerical Expressions with Radicals and rational exponents

Question 2

1.   √(64)
2.   ∛(-8)
3.   4 ^3/2
4.   8 ^2/3
5.   0.0001 ^3/4
6.   16 ^{- 3/4}
7.   √2 √8
8.   ∛2 ∛(32)
9.   √2 / √8
10.   ∛(-16) / ∛2
11.   - 2 (⁶√8) (5⁶√8)
12.   - 4 ∛(375) / (2∛3)

Simplify Algebraic Expressions with Exponents

Question 3

1.   x ² x ⁵
2.   y ⁴ / y ²
3.   (x ²) ^-2
4. 
   \( \large {3(\dfrac{1}{2} x^4)(\dfrac{1}{3} x^3) } \)
5. 
   \( \large {(2 x)^4(\dfrac{1}{4} x^{-3}) } \)
6. 
   \( \large { \dfrac{(3x)^2(-2x)^3}{(2x)^2} } \)
7. 
   \( \large { \dfrac{(4x)^2(100 x)^0}{(3x)^2} } \)
8.   (- 2 x² y ^-3)³
9.   (3 x² y³) (2 x⁵ y ^{- 2})
10.   (- 2 x² y³ z⁴) ( - 4 x³ y z ^{- 8}) ( 5 x y² z²)
11. 
    \( \large { (-2xy^2)^2 \left (\dfrac{x^6}{(2x)^2} \right)^3 } \)
12. 
    \( \large { \left (\dfrac{4 x^6 y^3}{-8x^3y^{-2}} \right) } \)
13. 
    \( \large { \left (\dfrac{- 4 x^3 y^2}{3 x^3 y^3} \right) \left (\dfrac{3 x^2 y^5}{-6x^3y^2} \right) } \)
14. 
    \( \large { \dfrac{1}{6}(x^2 y z^3)\left (\dfrac{4 x^2 y^3}{3 x^2 y^{-3}z^2} \right) \left (\dfrac{3 x^2 y^5 z}{-x^3 y^2 z^3} \right) } \)
15. 
    \( \large { (4 x^{2/3})(-8x^{1/2}) } \)
16. 
    \( \large { (4 x)^{3/2}(9x)^{1/2} } \)
17. 
    \( \large { (- 27 x^{3/2})^{1/3} } \)
18. 
    \( \large {\left (\dfrac{-8x^3}{y^{-6}} \right)^{2/3} } \)
19. 
    \( \large { x^{5/3} x^{1/6} x ^{-11/6} } \)

Simplify Algebraic Expressions with Radicals

Question 4

1. 
   \( \large { \sqrt{x^2} } \)
2. 
   \( \large { \sqrt[4]{16x^4} } \)
3. 
   \( \large { \sqrt{(2x - 1)^2} } \)
4. 
   \( \large { \sqrt[3]{-27x^3} } \)
5. 
   \( \large { \sqrt[3]{8 x^6 y^3} } \)
6. 
   \( \large { \sqrt{x^3} \sqrt{x^2} } \)
7. 
   \( \large { \sqrt[3]{\left (\dfrac{-8x^6}{y^{-3}} \right)} y} \)
8. 
   \( \large { \dfrac{ \sqrt[5]{64x^9 y^7}}{ \sqrt[5]{2 x^4 y^2}} } \)
9. 
   \( \large { (4\sqrt[8]{b^2})( 5\sqrt[8]{b^3})( \sqrt[8]{b^3}) } \)

Solutions to the Above Questions

Question 1

1.   3 ² = 3 × 3 = 9
2.   - 3 ⁴ = - (3 × 3 × 3 × 3 ) = - 81
3.   (- 3) ⁴ = - 3 × - 3 × - 3 × - 3 = 81
4.   12 ⁰ = 1
5.   0 ²⁰ = 0
6.   0 ^{- 4} = 1 / 0 ⁴ = 1 / 0 : undefined because division by zero is not allowed in maths.
7.   (-1) ⁷ = - 1
8.   (-1) ⁴ = 1
9.   (-1) ^{- 3} = 1 / (-1) ³ = 1 / (- 1) = - 1
10.   (- 2 / 3) ^-2 = (- 3 / 2) ² = (-1) ² ( 3 / 2) ² = 9 / 4
11.   (- 2) ^-2 / 4 ^-2 = 4 ² / (- 2) ² = 16 / 4 = 4
12.   - 3 ^-3 + (- 2) ^-2 = - 1 / 3 ³ + 1 / (- 2) ² = - 1 / 27 + 1 / 4 = 23 / 108
13.   x ⁴ for x = - 2
    x ⁴ = (-2) ⁴ = 16
14.   - x ² for x = - 3
    - x ² = - (-3) ² = - 9
15.   - x ⁵ for x = - 2
    - x ⁵ = -( - 2) ⁵ = 32
16.   x⁴ for x = - 2 / 3
    (- 2 / 3) ⁴ = (-2) ⁴ / (3) ⁴ = 16 / 81
17.   - x ^-2 for x = - 1 / 2
    - x ^-2 = - (- 1 / 2) ^-2 = - ( - 1) ^-2 ( 1 / 2) ^-2
    = - (1) (2 / 1) ² = - 4 / 1 = - 4
18.   - x ⁵ for x = - 1 / 3
    - x ⁵ = - ( - 1 / 3) ⁵ - ( - 1) ⁵( 1 / 3)⁵
    = - (-1) (1 / 243) = 1 / 243

Question 2

1.   √(64) = √(8 ²) = 8
2.   ∛(-8) = ∛((-2)³) = - 2
3.   4 ^3/2 = (√4) ³ = 2 ³ = 8
4.   8 ^2/3 = (∛(8)) ² = 2 ² = 4
5.   0.0001 ^3/4 = ( ⁴√(0.0001) ) ³
   = ( ⁴√(1/10000) ) ³ = ( ⁴√(1/10⁴) )³
   = ( 1 / 10 ) ³ = 1 / 10 ³ = 0.001
6.   16 ^{- 3/4} = 1 / 16 ^3/4 = 1 / ( ⁴√(16)) ³
   = 1 / 2 ³ = 1 / 8
7.   √2 √8 = √(2 × 8) = √(16) = 4
8.   ∛2 ∛(32) = ∛(2 × 32) = ∛(64) = 4
9.   √2 / √8 = √(2 / 8) = √(1 / 4) = 1 / 2
10.   ∛(-16) / ∛2 = ∛(-16/2) = ∛(- 8) = - 2
11.   - 2 ( ⁶√8) (5 ⁶√8) = (- 2 × 5) ⁶√(8 × 8)
    = - 10 ⁶√(64) = - 10 ⁶√(2⁶) = -10(2) = - 20
12.   - 4 ∛(375) / (2∛3) = (- 4 / 2) ∛ (375 / 3) = - 2 ∛ (375 / 3)
    = - 2 ∛ (125) = - 2 × 5 = - 10

Question 3

1.   x ² x ⁵ = x ^{2 + 5} = x⁷
2.   y ⁴ / y ² = y ^{4 - 2} = y ²
3.   (x ²) ^-2 = (x ^{2 × (-2)}) = x ^{- 4} = 1 / x ⁴
4. 
   \( \large {3(\dfrac{1}{2} x^4)(\dfrac{1}{3} x^3) = (3 \times \dfrac{1}{2} \times \dfrac{1}{3} ) x^{4 + 3} = \dfrac{1}{2} x^7} \)
5. 
   \( \large {(2 x)^4(\dfrac{1}{4} x^{-3}) = 2^4 x^4 (\dfrac{1}{4} x^{-3}) \\\\ = (2^4 \times \dfrac{1}{4}) x^{4-3} = 4 x} \)
6. 
   \( \large { \dfrac{(3x)^2(-2x)^3}{(2x)^2} = \dfrac{3^2 x^2 (-2)^3 x^3}{2^2 x^2} \\\\ = \dfrac{3^2 (-2)^3}{2^2} \dfrac{x^3}{x^2} = - 18 x } \)
7. 
   \( \large { \dfrac{(4x)^2(100 x)^0}{(3x)^2} = \dfrac{4^2 x^2 \times 1}{3^2 x^2} = 16/9 } \)
8.   (- 2 x ² y ^-3) ³ = (- 2) ³ (x ²) ³ (y ^-3) ³ = - 8 x ⁶ y ^-9 = - 8 x ⁶ / y ⁹
9.   (3 x² y ³) (2 x ⁵ y ^{- 2}) = (3 × 2)(x ² x ⁵)(y ³ y ^{- 2}) = 6 x ^{2 + 5} y ^{3 - 2} = 6 x ⁷ y
10.   (- 2 x ² y ³ z ⁴) ( - 4 x ³ y z ^{- 8}) ( 5 x y ² z ²) = (-2 × (-4) × 5)(x ² x ³ x)(y ³ y y ²)(z ⁴ z ^-8 z ²)
    = 40 x ^{2 + 3 + 1} y ^{3 + 1 + 2} z ^{4 - 8 + 2} = 40 x ⁶ y ⁶ z ^{- 2} = 40 x ⁶ y ⁶ / z ²
11. 
    \( \large { (-2xy^2)^2 \left (\dfrac{x^6}{(2x)^2} \right)^3 = (-2)^2 x^2 y^4 \left (\dfrac{x^{18}}{(2x)^6} \right) = \dfrac{4x^2 y^4x^{18}}{2^6 x^6}} \\\\ \large { = \dfrac{4}{2^6} \dfrac{x^{2+18}y^4}{x^6} = \dfrac{1}{16} {x^{14} y^4} } \)
12. 
    \( \large { \left (\dfrac{4 x^6 y^3}{-8x^3y^{-2}} \right) = \dfrac{4}{-8} \; \dfrac{x^6 y^3}{x^3y^{-2}} = -\dfrac{1}{2} x^{6-3} y^{3-(-2)} = -\dfrac{1}{2} x^3 y^5 } \)
13. 
    \( \large { \left (\dfrac{- 4 x^3 y^2}{3 x^3 y^3} \right) \left (\dfrac{3 x^2 y^5}{-6x^3y^2} \right) = \dfrac{(- 4 \times 3)}{(3 \times (-6))} \dfrac{(x^3 y^2)(x^2 y^5)}{(x^3 y^3)(x^3y^2)} }\\\\\\ \large { = \dfrac{2}{3} \dfrac{x^5 y^7}{x^6 y^5} = \dfrac{2}{3} x^{5-6} y^{7-5}= \dfrac{2}{3} \dfrac{y^2}{x} } \)
14. 
    \( \large { \dfrac{1}{6}(x^2 y z^3)\left (\dfrac{4 x^2 y^3}{3 x^2 y^{-3}z^2} \right) \left (\dfrac{3 x^2 y^5 z}{-x^3 y^2 z^3} \right) = \dfrac{1 \times 4 \times 3}{6 \times 3 \times (-1)} \dfrac{(x^2 y z^3)(x^2 y^3)(x^2 y^5 z)}{(x^2 y^{-3}z^2)(-x^3 y^2 z^3)} } \\\\\\ \large { = (-\dfrac{2}{3}) \dfrac{x^{2+2+2}y^{1+3+5}z^{3+1}}{x^{2+3} y^{-3+2} z^{2+3} } = - \dfrac{2}{3} \dfrac{x^6 y^9 z^4}{x^5 y z^5} = - \dfrac{2}{3} \; \dfrac{x y^{10} }{z} } \)
15. 
    \( \large { (4 x^{2/3})(-8x^{1/2}) = (4 \times; (-8))(x^{2/3+1/2}) = - 32 x^{7/6} } \)
16. 
    \( \large { (4 x)^{3/2}(9x)^{1/2} = 4^{3/2} x^{3/2} 9^{1/2} x^{1/2} = (\sqrt 4)^3 (\sqrt 9) x^{3/2+1/2} = 8 \times 3 x^2 = 24 x^2} \)
17. 
    \( \large { (- 27 x^{3/2})^{1/3} = (- 27)^{1/3} (x^{3/2})^{1/3} = \sqrt[3]{-27} x^{3/2 \times 1/3} = - 3 x^{1/2} = - 3 \sqrt x } \)
18. 
    \( \large {\left (\dfrac{-8x^3}{y^{-6}} \right)^{2/3} = \dfrac{(-8x^3)^{2/3}} {(y^{-6})^{2/3}} = \dfrac{(-8)^{2/3} (x^3)^{2/3}} {(y^{-6 \times (2/3)}}} \\\\\\ \large { = \dfrac{(\sqrt[3]{-8})^2 (x^{3 \times (2/3)})} {(y^{-4})} = 4 x^2 y^4 } \)
19. 
    \( \large { x^{5/3} x^{1/6} x ^{-11/6} = x^{5/3+1/6 -11/6} = x^0 = 1 , x \ne 0 } \)

Question 4

1. 
   \( \large { \sqrt{x^2} = | x |} \)
2. 
   \( \large { \sqrt[4]{16x^4} = \sqrt[4]{16} \sqrt[4]{x^4} = 4 | x | } \)
3. 
   \( \large { \sqrt{(2x - 1)^2} = |2x - 1| } \)
4. 
   \( \large { \sqrt[3]{-27x^3} = \sqrt[3]{-27} \sqrt[3]{x^3} = \sqrt[3]{(-3)^3} \sqrt[3]{x^3} = - 3 x } \)
5. 
   \( \large { \sqrt[3]{8 x^6 y^3} = \sqrt[3]{8} \sqrt[3]{ x^6} \sqrt[3]{y^3} = 2 x^{6/3} y = 2 x^2 y} \)
6. 
   \( \large { \sqrt{x^3} \sqrt{x^2} = \sqrt{x^3 x^2} = \sqrt{x^{3+2}} = \sqrt{x^5} = \sqrt{x^4 \times x} = x^2 \sqrt x } \)
7. 
   \( \large { \sqrt[3]{\left (\dfrac{-8x^6}{y^{-3}} \right)} = \dfrac{\sqrt[3]{-8x^6}}{\sqrt[3]{y^{-3}}} = \dfrac{\sqrt[3]{-8} \sqrt[3]{x^6}} {\sqrt[3]{y^{-3} }} }\\\\\\ \large { = \dfrac{\sqrt[3]{(-2)^3} \sqrt[3]{(x^2)^3}} {\sqrt[3]{(y^{-1})^3 }} = \dfrac{-2 x^2}{y^{-1}} = -2 x^2 y} \)
8. 
   \( \large { \dfrac{ \sqrt[5]{64x^9 y^7}}{ \sqrt[5]{2 x^4 y^2}} = \sqrt[5] {\dfrac{64x^9 y^7}{2 x^4 y^2}}} \\\\\\ \large { = \sqrt[5] {32 x^{9-4} y^{7-2}} = \sqrt[5] {2^5} \sqrt[5] x^{5} \sqrt[5] y^{5} = 2 x y } \)
9. 
   \( \large { (4\sqrt[8]{b^2})( 5\sqrt[8]{b^3})( \sqrt[8]{b^3}) = (4 \times 5) \sqrt[8]{b^2 b^3 b^3 } = 20 \sqrt[8]{b^8 } = 20 |b| = 20 b } \)
   Note: In the last question; since ⁸√(b³) is real then b is greater than or equal to zero. Hence |b| = b.

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