Evaluate the following expressions: a. √27, b. √81, c. √64, d. √1.
Understand the Problem
The question is asking to evaluate the given expressions involving square roots and cube roots based on the provided explanation about roots.
Answer
a. \( 3 \) b. \( 9 \) c. \( -4 \) d. \( 1 \)
Answer for screen readers
a. ( 3 )
b. ( 9 )
c. ( -4 )
d. ( 1 )
Steps to Solve
- Evaluate ( \sqrt[3]{27} )
Recognize that the cube root of 27 means we need to find a number that, when cubed, gives us 27.
We know that:
$$ 3^3 = 27 $$
Thus,
$$ \sqrt[3]{27} = 3 $$
- Evaluate ( \sqrt{81} )
To evaluate the square root of 81, we need to find a number that, when squared, results in 81.
We know that:
$$ 9^2 = 81 $$
Therefore,
$$ \sqrt{81} = 9 $$
- Evaluate ( \sqrt[3]{-64} )
For the cube root of -64, we need to find a number that, when cubed, equals -64.
We know that:
$$ (-4)^3 = -64 $$
So,
$$ \sqrt[3]{-64} = -4 $$
- Evaluate ( \sqrt{1} )
To find the square root of 1, we need a number that, when squared, equals 1.
We know that:
$$ 1^2 = 1 $$
Thus,
$$ \sqrt{1} = 1 $$
a. ( 3 )
b. ( 9 )
c. ( -4 )
d. ( 1 )
More Information
The cube root is the opposite of cubing a number, similar to square roots being the opposite of squaring a number. The expressions evaluated relate directly to finding numbers that satisfy ( x^2 ) or ( x^3 ).
Tips
- Confusing square and cube roots: Remember square roots deal with ( x^2 ) and cube roots with ( x^3 ).
- Overlooking negative numbers: The cube root of a negative number is negative, but the square root of a negative number is not defined in the real number system.
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