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Basically, parentheses are used in expressions to change the natural order of precedence among the operators. For example, consider this expression:

3 + 4 * 5

Since multiplication has precedence over addition, we know that this expression will be evaluated as follows: Four times five is twenty; twenty plus three is twenty-three. Or, in notation:

3 + 4 * 5

3 + 20

23

But what if you want to add the three and
the four ** first**, and

(3 + 4) * 5

One always evaluates the expression ** inside**
of parentheses

(3 + 4) * 5

7 * 5

35

** The expression inside of the parentheses
may have its own precedence for operations.** For example:

(2 + 4 * 6) * 3

Would work this way: Inside the parentheses multiplication has precedence over addition, so four would be multiplied by six to get twenty-four, and then two would be added to twenty-four to get twenty-six. That is the value inside of the parentheses. This value, twenty-six would then be multiplied by three to finally get a value of seventy-eight. In notation:

(2 + 4 * 6) * 3

(2 + 24) * 3

26 * 3

78

*Notice, regarding an example which we have
seen before:*

^{-}4^{2} = ^{-}16

That is, the negative of the second power of four is negative sixteen. But by using parentheses we can state:

(^{-}4)^{2} = 16

That is, the second power of negative four is positive sixteen, or negative four raised to the second power is positive sixteen.

*Here is another example from before:*

Notice how parentheses changes things; here, it is the third power of two which receives its value first, making a value of eight, and then this value is raised to the fourth power:

As we saw before, 2^{81} = 2,417,851,639,229,258,349,412,352.
But 8^{4} = 4,096. Certainly changing the natural
order of the operators with parentheses changes quite a bit
here.

** Sometimes parentheses have a use that does
not look much like they are being used to change the order of
operator precedence.** They can be used to

4 + (5 * 3)

However, they do no harm, and, perhaps, the author wants to show some special feature in a problem where the five times the three forms a unit of thought currently under discussion, so, it is emphasized this way.

Where are you?

Other Representations Besides Numbers for Values

Introduction to Operators and Operands

More about Operators and Operands

*Here: Parentheses*

And Now Including Variables and Functions

Just a Few Notes about Multiplication

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