Perfect Squares, Perfect Cubes, Square roots, Cube roots and Equations of the form x²=p and x^3=p
Objective:
Essential Question for Topic
Measurements frequently involve very large or very small numbers. How can you make such measurements easy to use and compare?
Focus Question for Lesson
How can you apply what you know about squares, cubes and square roots and cube roots to write and solve equations of the form x2 = p and x^3=p? How can you use equations in that form?
Key Vocabulary
perfect square, square root, cube root, perfect cubes
Standard
8.EEI.2�Evaluate the square root of a perfect square.�Evaluate the cube root of a perfect cube.�Solve equations of the form x^2 = p and x^3 = p where p is a positive rational number.�Approximate irrational solutions of equations of the form x^2 = p and x^3 = p where p is a positive rational number.�
Squaring a number means multiplying it by itself. So to square 3, we'd multiply 3 × 3, which we write as 32. While the answers to square roots can be both positive and negative (since either 22 or (-2)2 can equal 4).
To cube a number, just use it in a multiplication 3 times. (the little 3 means the number appears three times in multiplying. The cube root of 27 is ...
... 3, because when 3 is cubed you get 27.
The Square Root Property
If x2 = a, then x = √a or -√a.
The property above says that you can take the square root of both sides of an equation, but you have to think about two cases: the positive square root of a and the negative square root of a.
Square Root Of A Number
Positive square root: √16 = 4
Negative square root: -√ 16 = -4
Positive and negative square roots: ± √16= ±4
Negative numbers have no real roots: √-16
There is no real number that, when squared, would equal -16
This is a hint that equations involving x3 can have negative solutions.
Find all solutions to the Perfect Squares
Example 1:
x2 = 25
Solution: The square root of 25 is ±5.
Example 2:
x2 = 100: (±10)
Cube Root Property
To solve an equation of the form 𝘹^𝟥=𝘱, take the cube root of each side of the equation.
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When we solve equations, the solution sometimes requires finding a square or cube root of both sides of the equation.
When your equation simplifies to:
x2 = #
you must find the square root of both sides in order to find the value of x.
When your equation simplifies to:
x3 = #
you must find the cube root of both sides in order to find the value of x.
Try These
A
B
D
C
no real solution
A
B
D
C
no real solution
Square Roots of Decimals
Recall:
To find the square root of a decimal, convert the decimal to a fraction first. Follow your steps for square roots of fractions.
= .05
= .2
= .3
Evaluate
A
B
D
C
no real solution
Evaluate
A
C
D
B
.06
.6
6
No Real Solution
Example:
Solve.
Divide each side by the coefficient. Then take the square root of each side.
98
Solve.
100
Solve.