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The number
is between what two whole numbers?
Since
Since
Thus,
Thus,
is a number between 7 and 8.
Write the principal and secondary square roots of each number.
Use a calculator to obtain a decimal approximation for the two square roots of 35. Round to two decimal places.
Since we know that the square of any real number is a positive number or zero, we can see that expressions such as do not describe real numbers. There is no real number that can be squared that will produce −16. For to be a real number, we must have In our study of algebra, we will assume that all variables and all expressions in radicands represent nonnegative numbers (numbers greater than or equal to zero).
Write the proper restrictions that must be placed on the variable so that each expression represents a real number.
For
to be a real number, we must have
For
to be a real number, we must have
Write the proper restrictions that must be placed on the variable so that each expression represents a real number.
When variables occur in the radicand, we can often simplify the expression by removing the radical sign. We can do so by keeping in mind that the radicand is the square of some other expression. We can simplify a radical by seeking an expression whose square is the radicand. The following observations will help us find the square root of a variable quantity.
Since
is a square root of
Also
Since
is a square root of
Also
Since
is a square root of
Also
These examples suggest the following rule:
If a variable has an even exponent, its square root can be found by dividing that exponent by 2.
The examples of Sample Set B illustrate the use of this rule.
Simplify each expression by removing the radical sign. Assume each variable is
Simplify each expression by removing the radical sign. Assume each variable is nonnegative.
How many square roots does every positive real number have?
two
The symbol represents which square root of a number?
The symbol – represents which square root of a number?
secondary
For the following problems, find the two square roots of the given number.
For the following problems, evaluate each expression. If the expression does not represent a real number, write "not a real number."
For the following problems, write the proper restrictions that must be placed on the variable so that the expression represents a real number.
For the following problems, simplify each expression by removing the radical sign.
( [link] ) Find the sum.
( [link] ) Solve the equation, if possible:
No solution is excluded.
( [link] ) Perform the division:
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