Problem 12
Question
Evaluate each expression in Exercises \(1-12,\) or indicate that the root is not a real number. $$\sqrt{(-17)^{2}}$$
Step-by-Step Solution
Verified Answer
The result of evaluating the expression \(\sqrt{(-17)^{2}}\) is \(17\).
1Step 1: Evaluate the Inside Expression
First, calculate the value inside the square root, \((-17)^{2}\). When you square a negative number, the result is positive. So, \((-17)^{2} = 289\).
2Step 2: Evaluate the Square Root
Next, calculate the square root of the result from step 1, \(\sqrt{289}\). The square root of \(289\) is \(17\).
Key Concepts
Understanding Square RootsSquaring Negative NumbersProperties of Real NumbersBasic Mathematical Operations
Understanding Square Roots
A square root is a number that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\).
Square roots can be a bit tricky, especially when working with negative numbers. In the real number system, only non-negative numbers have real square roots, because multiplying two negative numbers yields a positive product. This means square roots of negative numbers (like \(\sqrt{-1}\)) are not real numbers.
Understanding the basics of square roots is key, especially when evaluating expressions. Simplifying the expression under the square root sign first can make solving it much easier.
Square roots can be a bit tricky, especially when working with negative numbers. In the real number system, only non-negative numbers have real square roots, because multiplying two negative numbers yields a positive product. This means square roots of negative numbers (like \(\sqrt{-1}\)) are not real numbers.
Understanding the basics of square roots is key, especially when evaluating expressions. Simplifying the expression under the square root sign first can make solving it much easier.
Squaring Negative Numbers
Squaring negative numbers can confuse some learners, so let's clarify it. When a negative number is squared, it becomes positive.
This is because multiplying the negative sign twice cancels itself out:
Think of it as multiplying the same negative number by itself. Since the negative sign appears twice, it results in a positive number. Remembering this principle can help avoid common mistakes.
This is because multiplying the negative sign twice cancels itself out:
- \((-3) \times (-3) = 9\)
- \((-17) \times (-17) = 289\)
Think of it as multiplying the same negative number by itself. Since the negative sign appears twice, it results in a positive number. Remembering this principle can help avoid common mistakes.
Properties of Real Numbers
Real numbers include all the numbers on the number line. This means both positive and negative integers, fractions, and irrational numbers. However, they do not include imaginary numbers.
Real numbers are very useful in evaluating expressions as they provide a complete set of values for most mathematical operations. When working with real numbers, remember that square roots of negative numbers are not included. Instead, such operations lead to results within the imaginary number set.
Exercises involving real numbers require understanding these properties to determine if the expressions are valid in the real number system.
Real numbers are very useful in evaluating expressions as they provide a complete set of values for most mathematical operations. When working with real numbers, remember that square roots of negative numbers are not included. Instead, such operations lead to results within the imaginary number set.
Exercises involving real numbers require understanding these properties to determine if the expressions are valid in the real number system.
Basic Mathematical Operations
Mathematical operations like addition, subtraction, multiplication, and division are fundamental.
These operations form the foundation of more complex operations, including powering and rooting. When evaluating expressions like \(\sqrt{(-17)^{2}}\), you first square the number and then find the square root, applying operations in the correct order ensures accuracy.
Following the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication & Division, Addition & Subtraction) helps systematically solve problems. Practicing these operations can make you more proficient in handling mathematical expressions.
Understanding these basics makes the process of evaluating mathematical expressions smooth and error-free.
These operations form the foundation of more complex operations, including powering and rooting. When evaluating expressions like \(\sqrt{(-17)^{2}}\), you first square the number and then find the square root, applying operations in the correct order ensures accuracy.
Following the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication & Division, Addition & Subtraction) helps systematically solve problems. Practicing these operations can make you more proficient in handling mathematical expressions.
- Multiply or divide before adding or subtracting.
- Apply exponents and roots in the right sequence to simplify expressions correctly.
Understanding these basics makes the process of evaluating mathematical expressions smooth and error-free.
Other exercises in this chapter
Problem 12
$$\text { Factor by grouping.}$$ $$x^{3}-3 x^{2}+4 x-12$$
View solution Problem 12
Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$\left(18 x^{4}-2 x^{3}-7 x+8\right)-\left(9 x^{4}-6
View solution Problem 13
Evaluate each algebraic expression for the given value or values of the variable(s). $$\frac{5(x+2)}{2 x-14}, \text { for } x=10$$
View solution Problem 13
Evaluate each exponential expression. $$2^{2} \cdot 2^{3}$$
View solution