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. \( \left( -\dfrac{2}{3} \right)^{-2} \)
11. \( \dfrac{(-2)^{-2}}{4^{-2}} \)
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 \( x = -\dfrac{2}{3} \)
17. \( -x^{-2} \) for \( x = -\dfrac{1}{2} \)
18. \( -x^5 \) for \( x = -\dfrac{1}{3} \)

Simplify Numerical Expressions with Radicals and rational exponents

Question 2

1. \( \sqrt{64} \)
2. \( \sqrt[3]{-8} \)
3. \( 4^{3/2} \)
4. \( 8^{2/3} \)
5. \( 0.0001^{3/4} \)
6. \( 16^{-3/4} \)
7. \( \sqrt{2} \sqrt{8} \)
8. \( \sqrt[3]{2} \sqrt[3]{32} \)
9. \( \dfrac{\sqrt{2}}{\sqrt{8}} \)
10. \( \dfrac{\sqrt[3]{-16}}{\sqrt[3]{2}} \)
11. \( -2 (\sqrt[6]{8})(5\sqrt[6]{8}) \)
12. \( \dfrac{-4 \sqrt[3]{375}}{2\sqrt[3]{3}} \)

Simplify Algebraic Expressions with Exponents

Question 3

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

Simplify Algebraic Expressions with Radicals

Question 4

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

Solutions to the Above Questions

Solution to Question 1

1. \(3^2 = 3 \times 3 = 9\)
2. \(-3^4 = - (3 \times 3 \times 3 \times 3) = -81\)
3. \((-3)^4 = -3 \times -3 \times -3 \times -3 = 81\)
4. \(12^0 = 1\)
5. \(0^{20} = 0\)
6. \(0^{-4} = \dfrac{1}{0^4} = \dfrac{1}{0}\) : undefined because division by zero is not allowed in maths.
7. \((-1)^7 = -1\)
8. \((-1)^4 = 1\)
9. \((-1)^{-3} = \dfrac{1}{(-1)^3} = \dfrac{1}{-1} = -1\)
10. \(\left(-\dfrac{2}{3}\right)^{-2} = \left(-\dfrac{3}{2}\right)^2 = (-1)^2 \left(\dfrac{3}{2}\right)^2 = \dfrac{9}{4}\)
11. \(\dfrac{(-2)^{-2}}{4^{-2}} = \dfrac{4^2}{(-2)^2} = \dfrac{16}{4} = 4\)
12. \(-3^{-3} + (-2)^{-2} = -\dfrac{1}{3^3} + \dfrac{1}{(-2)^2} = -\dfrac{1}{27} + \dfrac{1}{4} = \dfrac{23}{108}\)
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. \(x^4\) for \(x = -\dfrac{2}{3}\) \[ \left(-\dfrac{2}{3}\right)^4 = \dfrac{(-2)^4}{3^4} = \dfrac{16}{81} \]
17. \(-x^{-2}\) for \(x = -\dfrac{1}{2}\) \[ - x^{-2} = -\left(-\dfrac{1}{2}\right)^{-2} = -(-1)^{-2} \left(\dfrac{1}{2}\right)^{-2} \] \[ = -1 \cdot \left(\dfrac{2}{1}\right)^2 = -\dfrac{4}{1} = -4 \]
18. \(-x^5\) for \(x = -\dfrac{1}{3}\) \[ - x^5 = -\left(-\dfrac{1}{3}\right)^5 = -(-1)^5 \left(\dfrac{1}{3}\right)^5 \] \[ = -(-1) \cdot \dfrac{1}{243} = \dfrac{1}{243} \]

Solution to Question 2

1. \(\sqrt{64} = \sqrt{8^2} = 8\)
2. \(\sqrt[3]{-8} = \sqrt[3]{(-2)^3} = -2\)
3. \(4^{3/2} = (\sqrt{4})^3 = 2^3 = 8\)
4. \(8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4\)
5. \(0.0001^{3/4} = (\sqrt[4]{0.0001})^3\) \[ = (\sqrt[4]{\dfrac{1}{10000}})^3 = (\sqrt[4]{\dfrac{1}{10^4}})^3 \] \[ = \left(\dfrac{1}{10}\right)^3 = \dfrac{1}{10^3} = 0.001 \]
6. \(16^{-3/4} = \dfrac{1}{16^{3/4}} = \dfrac{1}{(\sqrt[4]{16})^3}\) \[ = \dfrac{1}{2^3} = \dfrac{1}{8} \]
7. \(\sqrt{2} \sqrt{8} = \sqrt{2 \times 8} = \sqrt{16} = 4\)
8. \(\sqrt[3]{2} \sqrt[3]{32} = \sqrt[3]{2 \times 32} = \sqrt[3]{64} = 4\)
9. \(\dfrac{\sqrt{2}}{\sqrt{8}} = \sqrt{\dfrac{2}{8}} = \sqrt{\dfrac{1}{4}} = \dfrac{1}{2}\)
10. \(\dfrac{\sqrt[3]{-16}}{\sqrt[3]{2}} = \sqrt[3]{\dfrac{-16}{2}} = \sqrt[3]{-8} = -2\)
11. \(-2 (\sqrt[6]{8})(5\sqrt[6]{8}) = (-2 \times 5)\sqrt[6]{8 \times 8}\) \[ = -10\sqrt[6]{64} = -10\sqrt[6]{2^6}\] \[ = -10(2) = -20 \]
12. \(\dfrac{-4\sqrt[3]{375}}{2\sqrt[3]{3}} = \left(\dfrac{-4}{2}\right)\sqrt[3]{\dfrac{375}{3}} = -2\sqrt[3]{125}\) \[ = -2 \times 5 = -10 \]

Solution to Question 3

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

Solution to Question 4

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

More References and links

• the rules for radicals and exponents
• simplify radical expressions
• Divide Radical Expressions
• Radical Expressions