Problem 57
Question
Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt{81}$$
Step-by-Step Solution
Verified Answer
The simplified expression is 9.
1Step 1: Identify the Expression
The given expression is \( \sqrt{81} \), which is the square root of 81.
2Step 2: Calculate the Square Root
Calculate \( \sqrt{81} \) by finding the number which, when multiplied by itself, equals 81. Knowing that \( 9 \times 9 = 81 \), we identify that \( \sqrt{81} = 9 \).
3Step 3: Verify and Conclude
Re-check the calculation by recalling the multiplication: \( 9\times9 = 81 \). Thus, \( \sqrt{81} = 9 \) is the correct simplification. Since \( \sqrt{81} \) has been simplified to a whole number, no further rationalization is needed.
Key Concepts
Square RootsRationalizationMathematical Expressions
Square Roots
Understanding square roots is a fundamental part of dealing with radicals. A square root asks the question: "What number, when multiplied by itself, gives me this number?" For example, if you take the square root of 81, you want to find a number that multiplied by itself results in 81.
The answer, in this case, is 9 because \( 9 \times 9 = 81 \). Knowing your multiplication tables can be very handy to quickly determine square roots.
Square roots can apply to perfect squares like 16, 25, 36, up to any greater number, simplifying them easily to 4, 5, and 6 respectively. Recognizing these perfect squares means recognizing numbers that are the products of integers squared.
The answer, in this case, is 9 because \( 9 \times 9 = 81 \). Knowing your multiplication tables can be very handy to quickly determine square roots.
Square roots can apply to perfect squares like 16, 25, 36, up to any greater number, simplifying them easily to 4, 5, and 6 respectively. Recognizing these perfect squares means recognizing numbers that are the products of integers squared.
- Perfect Squares: Numbers like 4, 9, 16, 25, 36, 49, etc.
- Using pairs of the same numbers in multiplication to find a square root.
- Quick recall through practice helps with automatic identification.
Rationalization
Rationalization is a technique used in mathematics mainly to get rid of radicals, or square roots, in the denominator of a fraction. This is usually preferable because having a radical in the denominator can make it more difficult to perform arithmetic calculations or further manipulations.
To rationalize, you multiply both the numerator and the denominator by a term that will remove the square root in the denominator. For example, if you have \( \frac{1}{\sqrt{2}} \), multiplying both the numerator and the denominator by \( \sqrt{2} \) results in \( \frac{\sqrt{2}}{\sqrt{4}} = \frac{\sqrt{2}}{2} \).
When the problem \( \sqrt{81} \) simplifies to 9, there is no denominator containing a square root to rationalize, so no further action is necessary. Rationalization generally follows these steps:
To rationalize, you multiply both the numerator and the denominator by a term that will remove the square root in the denominator. For example, if you have \( \frac{1}{\sqrt{2}} \), multiplying both the numerator and the denominator by \( \sqrt{2} \) results in \( \frac{\sqrt{2}}{\sqrt{4}} = \frac{\sqrt{2}}{2} \).
When the problem \( \sqrt{81} \) simplifies to 9, there is no denominator containing a square root to rationalize, so no further action is necessary. Rationalization generally follows these steps:
- Identify a fraction with a square root in the denominator.
- Multiply by the appropriate conjugate or radical to eliminate the square root.
- Simplify the fraction where possible.
Mathematical Expressions
Every mathematical expression is a combination of numbers, variables, and operations like addition, subtraction, multiplication, or division. When you encounter a radical sign like \( \sqrt{} \), it represents a square root operation within such an expression.
Expressions can be broken down and manipulated using various arithmetic rules and order of operations, ensuring that each step performed does not change the expression's value.
Simplifying expressions usually involves reducing to the simplest form or terms that are easier to comprehend. In the given exercise of \( \sqrt{81} \), it simplifies directly to the whole number 9. This simplification is often desirable for easier handling and further computation when solving equations or inequalities.
Importantly, ensuring clarity in expressions supports identifying and using algebraic structures effectively.
Expressions can be broken down and manipulated using various arithmetic rules and order of operations, ensuring that each step performed does not change the expression's value.
Simplifying expressions usually involves reducing to the simplest form or terms that are easier to comprehend. In the given exercise of \( \sqrt{81} \), it simplifies directly to the whole number 9. This simplification is often desirable for easier handling and further computation when solving equations or inequalities.
Importantly, ensuring clarity in expressions supports identifying and using algebraic structures effectively.
- Consistent application of arithmetic operations.
- Breaking expressions into simpler terms or components.
- Using variables alongside numbers in expressions for equations.
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