Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)³ = 5(1/3)³ to hold, so (51/3)³ must equal 5.
Complete the table. Give the area of each square in square units, and give its exact side length in units.
figure
A
B
C
D
E
area
1
side length
1
This table includes areas in square units and side lengths in units of some more squares. Complete the table.
area
9
23
89
side length
4
6.4
2.3
Activity
Cube It
Standards Alignment
Building On
8.EE.2
Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
A cube an has edge length of 3 units. What is the volume of the cube?
A cube has an edge length of 4 units. What is the volume of the cube?
A cube has a volume of 8 units. What is the edge length of the cube?
A cube has a volume of 7 units. What is the edge length of the cube?
is between 10 and 11 because and . Determine the whole numbers that each of these cube roots lies between:
Student Lesson Summary
If a square has a side length of , then the area is . If a square has an area of , then the side length is . For a positive number , the square root of is defined as the positive number that squares to make , and it is written as . In other words, . We can also think of as a solution to the equation . This square has an area of because its sides have length :
Similarly, if a cube has an edge length of , then the volume is . If a cube has a volume of , then the edge length is . The number is defined as the number that cubes to make . In other words, . We can also think of as a solution to the equation . This cube has a volume of because its sides have a length of :
Glossary
None
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Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)³ = 5(1/3)³ to hold, so (51/3)³ must equal 5.
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)³ = 5(1/3)³ to hold, so (51/3)³ must equal 5.