Simplify Exponents and Radicals

Practice Questions, including challenging ones, with Step-by-Step Solutions

To simplify exponents and radicals, you must apply the laws of exponents and the properties of roots. Review the rules for radicals and exponents before starting. Try to solve these numerical expressions without a calculator.

Simplify Numerical Expressions with Exponents

Question 1: Evaluate the following expressions:

  1. \( 3^2 \)
  2. \( -3^4 \)
  3. \( (-3)^4 \)
  4. \( \left( -\dfrac{2}{3} \right)^{-2} \)
  5. \( -3^{-3} + (-2)^{-2} \)
Show Solutions
  1. \( 3^2 = 9 \)
  2. \( -3^4 = -(3 \times 3 \times 3 \times 3) = -81 \)
  3. \( (-3)^4 = -3 \times -3 \times -3 \times -3 = 81 \)
  4. \( \left(-\dfrac{2}{3}\right)^{-2} = \left(-\dfrac{3}{2}\right)^2 = \dfrac{9}{4} \)
  5. \( -3^{-3} + (-2)^{-2} = -\dfrac{1}{27} + \dfrac{1}{4} = \dfrac{-4 + 27}{108} = \dfrac{23}{108} \)

Numerical Expressions with Radicals & Rational Exponents

Question 2: Evaluate the following:

  1. \( \sqrt[3]{-8} \)
  2. \( 8^{2/3} \)
  3. \( 16^{-3/4} \)
  4. \( \dfrac{\sqrt[3]{-16}}{\sqrt[3]{2}} \)
Show Solutions
  1. \( \sqrt[3]{-8} = \sqrt[3]{(-2)^3} = -2 \)
  2. \( 8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4 \)
  3. \( 16^{-3/4} = \dfrac{1}{(\sqrt[4]{16})^3} = \dfrac{1}{2^3} = \dfrac{1}{8} \)
  4. \( \dfrac{\sqrt[3]{-16}}{\sqrt[3]{2}} = \sqrt[3]{\dfrac{-16}{2}} = \sqrt[3]{-8} = -2 \)

Simplify Algebraic Expressions with Exponents

Question 3: Simplify these expressions:

  1. \( (x^2)^{-2} \)
  2. \( \dfrac{(3x)^2(-2x)^3}{(2x)^2} \)
  3. \( (-2x^2 y^{-3})^3 \)
  4. \( \left (\dfrac{-8x^3}{y^{-6}} \right)^{2/3} \)
Show Solutions
  1. \( (x^2)^{-2} = x^{-4} = \dfrac{1}{x^4} \)
  2. \( \dfrac{9x^2 \cdot (-8x^3)}{4x^2} = \dfrac{-72x^5}{4x^2} = -18x^3 \)
  3. \( (-2)^3 (x^2)^3 (y^{-3})^3 = -8x^6 y^{-9} = \dfrac{-8x^6}{y^9} \)
  4. \( \dfrac{(-8)^{2/3} (x^3)^{2/3}}{(y^{-6})^{2/3}} = \dfrac{4x^2}{y^{-4}} = 4x^2 y^4 \)

Simplify Algebraic Expressions with Radicals

Question 4: Simplify the following:

  1. \( \sqrt[4]{16x^4} \)
  2. \( \sqrt[3]{8 x^6 y^3} \)
  3. \( \dfrac{ \sqrt[5]{64x^9 y^7}}{ \sqrt[5]{2 x^4 y^2}} \)
Show Solutions
  1. \( \sqrt[4]{16} \sqrt[4]{x^4} = 2|x| \)
  2. \( 2x^{6/3}y^{3/3} = 2x^2y \)
  3. \( \sqrt[5]{\dfrac{64x^9 y^7}{2x^4 y^2}} = \sqrt[5]{32x^5 y^5} = 2xy \)

Challenge Questions

Question 5: Simplify the following advanced expressions. Assume all variables represent positive real numbers.

  1. \( \sqrt{x \sqrt{x \sqrt{x}}} \)
  2. \( \left( \dfrac{81 x^{-4} y^8}{16 x^8 y^{-4}} \right)^{-3/4} \)
  3. \( \dfrac{ \sqrt[3]{a^2b} \cdot \sqrt{ab^3} }{ \sqrt[6]{a^5b^7} } \)
Show Solutions

  1. Step 1: Convert innermost radicals to fractional exponents.
    \( \sqrt{x \sqrt{x \cdot x^{1/2}}} = \sqrt{x \sqrt{x^{3/2}}} \)

    Step 2: Continue converting outwards by multiplying powers.
    \( \sqrt{x \cdot (x^{3/2})^{1/2}} = \sqrt{x \cdot x^{3/4}} = \sqrt{x^{7/4}} \)

    Step 3: Final conversion.
    \( (x^{7/4})^{1/2} = \mathbf{x^{7/8}} \)

  2. Step 1: Simplify the fraction inside the parentheses first. Subtract the denominator's exponents from the numerator's.
    \( \dfrac{81}{16} x^{-4-8} y^{8-(-4)} = \dfrac{81}{16} x^{-12} y^{12} \)

    Step 2: Apply the negative sign from the fractional exponent, which flips the fraction.
    \( \left( \dfrac{16 x^{12}}{81 y^{12}} \right)^{3/4} \)

    Step 3: Apply the \(3/4\) power to each term (noting that \(16 = 2^4\) and \(81 = 3^4\)).
    \( \dfrac{(2^4)^{3/4} (x^{12})^{3/4}}{(3^4)^{3/4} (y^{12})^{3/4}} = \mathbf{\dfrac{8 x^9}{27 y^9}} \)

  3. Step 1: Convert all radicals to rational exponents.
    Numerator: \( (a^2b)^{1/3} \cdot (ab^3)^{1/2} = a^{2/3}b^{1/3} \cdot a^{1/2}b^{3/2} \)
    Denominator: \( (a^5b^7)^{1/6} = a^{5/6}b^{7/6} \)

    Step 2: Combine the terms in the numerator by adding their exponents.
    \( a^{2/3 + 1/2} b^{1/3 + 3/2} = a^{7/6} b^{11/6} \)

    Step 3: Divide by subtracting the denominator's exponents.
    \( a^{7/6 - 5/6} b^{11/6 - 7/6} = a^{2/6} b^{4/6} \)

    Step 4: Simplify the fractions and rewrite as a radical.
    \( a^{1/3} b^{2/3} = \mathbf{\sqrt[3]{ab^2}} \)

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