Square Rooting Equations

Slideshow 19,Mathematics,Mr Richard Sasaki,Room 307

Objectives• Recall how to square root a number• Understanding order of operations

and why we often square root last• Solve equations where we need to

square root and cube root

Square RootingWhat is the square root of ?Well . Because . Is that all?There is another square root of . What other number multiplies by itself to make ?−5 so is also the square root of .

When we square root a number, we get two solutions (unless we square root zero).

Note: The symbol is considered to have solution in some countries but 2 in others.

Square UnknownsIf an equation has a square unknown (like ), then the equation is non-linear. To solve an equation with a square unknown, we must square root it.Example

Solve .

√❑ √❑We must square root both sides.

𝑥=±9Note: If you prefer, you can write or . is simply more compact.

Cubic UnknownsCubic unknowns or unknowns to powers other than or in equations make equations non-linear too.Example

Solve .3√❑

We must cube root both sides.

𝑥=5Why isn’t the answer ?

Well . So the only solution is 5.The inverse of , is also non-linear.

Answers - Easy

𝑥=±2𝑥=±3𝑥=±6𝑥=0𝑥=±7𝑥=±8𝑥=±10𝑥=±9𝑥=±20𝑥=±12𝑥=±25𝑥=±

12

𝑥=2𝑥=1 𝑥=3𝑥=10𝑥=4 𝑥=7

Answers - Hard

𝑥=12 𝑥=

13𝑥=

16𝑥=0.5 𝑥=0.1 𝑥=9

𝑥=2 𝑥=3 𝑥=5 is always equal to no matter the value of so could be anything.

𝑥=13𝑥=4 𝑥=

15

𝑥=14𝑥=3 𝑥=

15

. As we cannot divide by , .

Order of OperationsAs hopefully most of you have realized. We apply the order of operations in reverse to solve equations! This means we deal with powers last (except when we have brackets). ExampleSolve .

Note: Remember, in general!−42 𝑥2=50

÷2𝑥2=25√❑𝑥=±5You should keep a neat layout!

Answers - Easy

𝑥2=9𝑥=±3

𝑥2=4𝑥=±2

𝑥2=25𝑥=±5

𝑥2=16𝑥=± 4

𝑥2=36𝑥=±6

𝑥2=81𝑥=±9

𝑥2=49𝑥=±7

𝑥2=49𝑥=±7

2 𝑥2=18𝑥2=9𝑥=±3

4 𝑥2=16𝑥2=4𝑥=±2

3 𝑥2=3𝑥2=1𝑥=±1

4 𝑥2=1𝑥2=

14

𝑥=±12

Answers – Hard (Top)𝑥4=16𝑥=±2

𝑥3=64𝑥=4 𝑥3=125

𝑥=5𝑥+1=±6

𝑥=−7 𝑜𝑟 5𝑥−2=±7

𝑥=−5𝑜𝑟 92 𝑥+3=±13

2 𝑥=−16𝑜𝑟 10𝑥=−8𝑜𝑟 5

2 𝑥3=16𝑥3=8𝑥=2

𝑥3

2=108

𝑥3=216𝑥=6

3 𝑥−1=113 𝑥=12𝑥=4

Answers – Hard (Bottom)

No real number multiplied by itself is equal to (or in fact, any negative number).

𝑥=−1

must be an odd number ( or a negative odd number (. Only odd number powers can produce negative results.