This section is rated INTERMEDIATE

An equation may contain a quantity raised to a power. Again, so long as the same operation is performed on both sides, the equation can be transformed to solve for an unknown variable. Taking a very simple example:

*x*^{2} = 25

√*x*^{2} = √25

*x* = 5

The symbol √ indicates the square root of a number - note that the square root can also be written as *x*^{0.5}. You need to take care when dealing with other quantities in the equation, so that these are transformed first and leave the power as the final transformation. Thus:

3*x*^{2} + 7 = 19

First shift the offset, +7:

3*x*^{2} = 19 - 7

Then deal with the multiplier, 3:

*x*^{2} = (19 - 7)/3 = 12/3 = 4

Finally, use square root of each side to solve for *x*:

*x*^{2} = 4

√*x*^{2} = √4

*x* = 2

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If you try to do these operations in a different order, you will soon get into a rather complex calculation where there is plenty of scope for making mistakes.

You can solve equations involving higher powers in the same way, provided that there is only one power in the equation or the equation can be transformed to yield a single power. However, if you are faced by an equation that contains more than one power you will need to use a formula to solve it. For instance, you cannot use the methods described here to solve this equation:

2*x*^{2} + 3*x* = 29