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Unit 10: Exponents and Radicals

The exponent rules turn repeated multiplication into fast arithmetic, and radicals are just exponents written the other way around.

6 topics · 20 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Exponent Rules: Product and Quotient

  1. How it worksHow does writing out the repeated multiplication behind x3 times x4 prove the product rule?
  2. Worked exampleFind the simplified form of x5 times x2, divided by x3.
  3. Common mistakeWhy doesn't x3 times x4 equal x to the 12th power?
Your turn1 of 2
Simplify x6 times x5, divided by x3.

Power of a Power and Power of a Product

  1. How it worksHow does a power applied to a product distribute onto every factor inside it?
  2. Worked exampleFind the simplified form of the quantity 2x3, raised to the 4th power.
  3. Common mistakeWhy doesn't the quantity x3, raised to the 4th power, give the same result as x3 times x4?
Your turn1 of 2
Simplify the quantity 4x3, raised to the 3 power.

Zero and Negative Exponents

  1. How it worksHow does extending the pattern of dividing by x each time force x0 to equal 1 and x−1 to equal a fraction?
  2. Worked exampleFind the simplified form of 3x−2 times x5.
  3. Common mistakeWhy doesn't a negative exponent make a number's value negative?
  4. Going furtherWhat happens to a fraction with a negative exponent when you flip it upside down?
Your turn1 of 2
Simplify 7x−3 times x9.

Simplifying Radical Expressions

  1. How it worksHow does pulling out the largest perfect-square factor simplify a square root?
  2. Worked exampleFind the simplified form of the square root of 72.
  3. Common mistakeWhy isn't the square root of a sum the same as the sum of the square roots?
Your turn1 of 2
Write the square root of 11 times x2, for x > 0, in simplest form, as a function of x.

Rational Exponents

  1. How it worksHow does the denominator of a rational exponent become the root and the numerator become the power?
  2. Worked exampleFind the value of 8 to the 2/3 power.
  3. Ties togetherHow does a rational exponent connect the exponent rules to the rules for simplifying radicals?
Your turn1 of 2
Simplify (x2) raised to the 2/2 power, for x > 0, as a function of x.

Operations with Radicals

  1. Worked exampleFind the sum of 3 times the square root of 2, and 5 times the square root of 2.
  2. Worked exampleFind the product of the square root of 12 and the square root of 3.
  3. Common mistakeWhy doesn't the square root of 3 plus the square root of 12 stay as two separate unsimplified terms?
  4. Worked exampleFind the quotient of the square root of 20 and the square root of 5.
Your turn1 of 2
The square root of 5 and a second radical that simplifies to 3 times the square root of 5 appear in an expression together. Can they be combined into a single term?