Equations

Ontario 09 Academic (MPM1D)

Squares and cubes in equations

Lesson

As we have already learnt, backtracking is a great way of solving equations. The backtracking process involves working backwards through the changes that have been made to a value by using inverse (opposite) operations.

So, if our questions involve square numbers or cube numbers, we can use their inverse operations (square roots and cube roots). to find the original unknown value.

The picture below shows the inverse relationship between a squared number and its square root. When $3$3 is squared ($3\times3$3×3), the result is $9$9. When backtracking, the square root of $9$9, is $3$3.

Watch out!

There are actually two answers when we find the square root of a number- a positive value and a negative value.

This is because the product of two negative numbers is a positive number. So squaring a negative number will give a positive answer e.g. $x\times x=x^2$`x`×`x`=`x`2 but $\left(-x\right)\times\left(-x\right)=x^2$(−`x`)×(−`x`)=`x`2. So the solution to $\sqrt{x^2}$√`x`2 is $x$`x` and $-x$−`x`.

This is why we can't have the square root of a negative number.

As we have just discussed, there are two answers when we find the square root of a number- one positive and one negative. We know mathematicians love to use shorthand abbreviations whenever possible. In this case, mathematicians use the plus minus [+-] symbol to show that there is a positive and a negative answer. For example,when finding the solution to $x^2=25$`x`2=25, instead of writing $x=5$`x`=5 or $-5$−5, we can simply write $x=\pm5$`x`=±5,

The process is the same for cubed numbers. The picture below shows that when $x$`x` is multiplied by itself twice, the result is $x^3$`x`3. Conversely, $\sqrt[3]{x^3}$^{3}√`x`3 equals $x$`x`.

Remember!

When backtracing to solve equations with square or cube numbers, we still need to follow the reverse order of operations.

Square roots and cubed roots can give rational or irrational answer. Rational answers will be nice whole numbers or numbers that can be expressed as a fraction. Irrational numbers will be decimals that cannot be written as fractions. Unless we're told otherwise, we want to give exact answers. If our answer is irrational, we should leave our answer as a radical.

Solve $x^2=121$`x`2=121.

Solve $x^3=64$`x`3=64.

Solve $x^2-15=100$`x`2−15=100, rounding your solutions to two decimal places.

Are the exact solutions to the equation rational or irrational values?

Rational

AIrrational

BRational

AIrrational

B

Solve $\frac{x^2}{6}=\frac{49}{6}$`x`26=496.

- Is your answer rational or irrational?
Rational

AIrrational

BRational

AIrrational

B

Relate their understanding of inverse operations to squaring and taking the square root, and apply inverse operations to simplify expressions and solve equations