pAIrmatic Lessons
UNIT 1Exponents and radicalsLesson 1 of 21 Rational Exponents2 Simplifying Radicals

Rational Exponents

A1.5Algebra 1Functions20 practice questions

An exponent can be a fraction. The bottom number is a root. The top number is a power.

What you'll be able to do

The big idea

The bottom number is a root. 64¹ᐟ³ is the cube root of 64. It is 4, because 4³ = 64.

The top number is a power. 1,000²ᐟ³ means take the cube root, then square it. The cube root of 1,000 is 10, and 10² = 100.

Root first. You can do the root or the power first. The root first keeps the numbers small.

A minus sign means flip. 9⁻³ᐟ² = 1 ÷ 9³ᐟ². Since 9³ᐟ² = 3³ = 27, the value is 1/27. It is not negative.

The old rules still work. To multiply, add the exponents: x¹ᐟ² · x¹ᐟ⁴ = x³ᐟ⁴. For a power of a power, multiply them: (x⁶)²ᐟ³ = x⁴.

Undo a power with its flip. To solve x³ᐟ² = 1,000, raise both sides to the 2/3 power. Then x = 1,000²ᐟ³ = 100.

Odd roots of negatives are fine. (−27)¹ᐟ³ = −3, because (−3)³ = −27. But (−9)¹ᐟ² has no real value. No real number squared is negative.

Words you need

Exponent
The small raised number. In 32³ᐟ⁵, the exponent is 3/5.
Base
The number being raised to a power. In 32³ᐟ⁵, the base is 32.
Power
A base with an exponent, like 2⁵. It also means the exponent itself.
Rational exponent
An exponent that is a fraction, like 2/3.
Square root
A number that gives the first number when you multiply it by itself. √25 = 5.
Cube root
A number you use 3 times as a factor to get the first number. ∛8 = 2.
Root
ⁿ√a is a number you use n times as a factor to get a. ⁴√81 = 3.
Index
The small number on a root sign. In ⁴√81, the index is 4.
Radical
A root sign and what is under it, like √5 or ∛x.
Factor
A number you multiply by. In 2 · 2 · 2, 2 is a factor.
Product
The answer when you multiply.
Reciprocal
1 divided by a number. The reciprocal of 8 is 1/8.
Variable
A letter that stands for a number, like x.
Expression
Numbers and letters joined by +, −, × or ÷. It has no = sign.
Simplify
Write it in a shorter form with the same value.
Coefficient
The number in front of a letter. In 70m, it is 70.
Constant
A number that does not change, like 70 in E = 70m³ᐟ⁴.
Logarithm
The power a base needs to make a number. log₂(8) = 3, because 2³ = 8.

Exponent form and root form

Exponent formRoot formValue
16¹ᐟ²√164
64¹ᐟ³∛644
81¹ᐟ⁴⁴√813
1,000²ᐟ³(∛1,000)²100
9⁻¹ᐟ²1 ÷ √91/3

Worked examples

Example 1

Find 32³ᐟ⁵ and 243⁻²ᐟ⁵ without a calculator.

  1. The 5 on the bottom means the 5th root. ⁵√32 = 2, because 2⁵ = 32.
  2. The 3 on top means cube it. 2³ = 8. So 32³ᐟ⁵ = 8.
  3. For 243⁻²ᐟ⁵, flip first. It is 1 ÷ 243²ᐟ⁵.
  4. ⁵√243 = 3, and 3² = 9. So 243⁻²ᐟ⁵ = 1/9.

Answer: 32³ᐟ⁵ = 8 and 243⁻²ᐟ⁵ = 1/9

Example 2

Simplify (64x⁶y⁻³)²ᐟ³. Write the answer with positive exponents. All variables are positive.

  1. Raise each part to the 2/3 power.
  2. 64²ᐟ³ = (∛64)² = 4² = 16.
  3. Multiply the exponents: (x⁶)²ᐟ³ = x⁴ and (y⁻³)²ᐟ³ = y⁻².
  4. Move y⁻² to the bottom to make its exponent positive. The answer is 16x⁴/y².

Answer: 16x⁴/y²

Example 3

Solve 2x³ᐟ² + 5 = 59.

  1. Subtract 5 from both sides: 2x³ᐟ² = 54.
  2. Divide both sides by 2: x³ᐟ² = 27.
  3. Raise both sides to the flip power, 2/3: x = 27²ᐟ³.
  4. 27²ᐟ³ = (∛27)² = 3² = 9. Check: 9³ᐟ² = 3³ = 27.

Answer: x = 9

Common mistakes

Multiplying by the exponent. 64²ᐟ³ is not 64 × 2/3. Take the cube root, then square it: 64²ᐟ³ = 16.

Making the answer negative. A minus sign in the exponent means flip. 81⁻¹ᐟ⁴ = 1/3, not −3.

Flipping the fraction. The root goes on the bottom. ∛(x⁴) = x⁴ᐟ³, not x³ᐟ⁴.

Adding exponents for a power of a power. (x²ᐟ³)³ = x², not x¹¹ᐟ³. Multiply the exponents.

Practice: 20 questions

11 of 20 are word problems. For a word problem, sketch it first and mark what you know.

Tier 1 Foundations Use the rule directly

  1. Rewrite ⁵√(x³) and 1 / ⁴√(y⁷) with fractional exponents, and rewrite a³ᐟ² and (8m)¹ᐟ³ in radical form. Simplify any numerical roots. All variables are positive.
  2. A cube-shaped shipping crate has a volume of 8 cubic feet. The area A of one square face of a cube with volume V is A = V²ᐟ³. Tenzin calculates the area of one face as 8²ᐟ³ = 16/3 square feet. Find his mistake and give the correct area in square feet. Word problem
  3. Without a calculator, find the exact value of 27²ᐟ³ and the exact value of 16³ᐟ⁴. Which expression is greater, and by how much?
  4. Simplify a¹ᐟ² · a¹ᐟ³ and b⁷ᐟ⁴ ÷ b¹ᐟ⁴. Write your answers with positive exponents. All variables are positive.
  5. Biologists model a mammal's resting energy use with the formula E = 70m³ᐟ⁴, where m is the animal's mass in kilograms and E is the energy used in kilocalories per day. Kwesi weighs a dog whose mass is 16 kilograms. How many kilocalories per day does the dog use at rest? Word problem

Tier 2 Intermediate One more step to think through

  1. A mammal's resting heart rate R, in beats per minute, is modeled by R = 240m⁻¹ᐟ⁴, where m is the animal's mass in kilograms. Marisol checks a horse with a mass of 625 kilograms. What is the horse's predicted resting heart rate in beats per minute? Word problem
  2. Simplify (16x⁸y⁻⁴)³ᐟ⁴. Write your answer with positive exponents only. All variables are positive.
  3. Nikolai simplifies 8⁻¹ᐟ³ · (x¹ᐟ²)³ and writes −2x⁷ᐟ². He made one mistake on the number and one mistake on the variable. Find both mistakes and write the correct simplified expression with positive exponents. Assume x is positive.
  4. For a planet orbiting a star at an average distance of d astronomical units, the length of one orbital year is T = d³ᐟ² Earth years. Planet A orbits at an average distance of d = 4 astronomical units. Planet B orbits at d = 9 astronomical units. Zayd compares their orbits. Which planet has the longer orbital year, and by how many Earth years? Word problem
  5. Solve the equation 2x³ᐟ⁴ − 5 = 11 for x.

Tier 3 Advanced Several steps - find the hidden structure

  1. A mammal's resting energy use E, in kilocalories per day, is modeled by E = 70m³ᐟ⁴, where m is the animal's mass in kilograms. A zoo animal uses 1,890 kilocalories per day at rest. What is the animal's mass in kilograms? Word problem
  2. Haruki orders 32⁻²ᐟ⁵, 27⁻¹ᐟ³, and 125⁻¹ᐟ³ from least to greatest without a calculator. Find the exact fraction for each expression and write the three original expressions in order from least to greatest.
  3. The total surface area S, in square centimeters, of a cube with volume V cubic centimeters is S = 6V²ᐟ³. Astrid molds a clay cube with a volume of 125 cubic centimeters. She then molds a second clay cube whose total surface area is 384 square centimeters. What is the total surface area of the first cube in square centimeters, and what is the volume of the second cube in cubic centimeters? Word problem
  4. Simplify (27x⁹ᐟ⁴y⁻³)¹ᐟ³ ÷ (x¹ᐟ⁴y⁻²). Write your answer with positive exponents only. All variables are positive.
  5. Pranav types (−8)¹ᐟ³ and (−4)¹ᐟ² into his graphing calculator. It shows −2 for the first one and an error for the second one. Pranav says the calculator is wrong about the first one, because a negative number can't have a fractional power. Find his mistake. Explain why the first has a real value and the second has none. Word problem

Tier 4 Challenge Stretch problems, some above grade level

  1. Define x²ᐟ³ as (∛x)². When solving the equation x²ᐟ³ = 9, a student raises both sides to the 3/2 power, gets x = 9³ᐟ² = 27, and says 27 is the only real solution. Find what the student missed and give all real solutions to the equation.
  2. Biologists use E = 70m³ᐟ⁴ to model a mammal's daily resting energy E, in kilocalories, from its mass m in kilograms. A capybara has 16 times the mass of a guinea pig. Chiamaka compares both their total daily energy use and their energy use per kilogram of body mass, which is E ÷ m. How many times as much total daily energy does the capybara use as the guinea pig? Which animal uses more energy per kilogram of its own body mass, and how many times as much? Word problem
  3. Mateo writes the orbit rule for planets as T = d⁹ᐟ⁴ ÷ ⁴√(d³). Here d is a planet's average distance from the Sun in astronomical units, and T is its year in Earth years. Simplify his rule to a single fractional power of d. A planet orbits at d = 9. How many times as long as Earth's year (d = 1) is its year? Word problem
  4. A microbiologist studies a cell culture across two growth phases. During Phase 1, the number of cells is multiplied by 8¹ᐟ². During Phase 2, the number of cells is multiplied by 4³ᐟ⁴. Over both phases combined, the starting population is multiplied by 2ᵏ. Find the exact value of k, and state how many times as large the cell population is after both phases together. Word problem
  5. A logarithm tells you a power. log₁₀(1,000) = 3 because 10³ = 1,000. Sound is measured in decibels. A sound that is I times as strong as the softest sound you can hear has a loudness of 10 · log₁₀(I) decibels. A fan is 10,000³ᐟ⁴ times as strong as the softest sound. A street is 100,000⁴ᐟ⁵ times as strong. How loud is each sound in decibels? How many decibels louder is the street? Word problem Above grade level

Answers

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