Problem 34
Question
Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{\sqrt{-25}}{\sqrt{-16} \sqrt{-81}} $$
Step-by-Step Solution
Verified Answer
The expression is \(0 - \frac{5}{36}i\).
1Step 1: Simplify the Square Roots of Negative Numbers
The square root of a negative number can be expressed in terms of the imaginary unit, denoted as \(i\), where \(i^2 = -1\). Thus, the square root of \(-25\) is \(5i\), the square root of \(-16\) is \(4i\), and the square root of \(-81\) is \(9i\).
2Step 2: Substitute the Expressions with Imaginary Numbers
Substitute the simplified square roots back into the original expression:\[\frac{5i}{4i \times 9i}.\]
3Step 3: Simplify the Denominator
Multiply the imaginary numbers in the denominator:\[4i \times 9i = 36i^2.\]Remember that \(i^2 = -1\), so \[36i^2 = 36(-1) = -36.\]
4Step 4: Divide the Numerator by the Denominator
Now divide the numerator by the simplified denominator:\[\frac{5i}{-36}.\]This expression can be rewritten as:\[-\frac{5}{36}i.\]
5Step 5: Write the Expression in the Form \(a + bi\)
The expression \(-\frac{5}{36}i\) can be written as \(a + bi\), where \(a = 0\) and \(b = -\frac{5}{36}\). Therefore, the expression is:\[0 - \frac{5}{36}i.\]
Key Concepts
Imaginary UnitSimplifying ExpressionsSquare Roots of Negative Numbers
Imaginary Unit
In mathematics, the imaginary unit is a fundamental concept used in complex numbers. It is denoted by the symbol \(i\), and it is the foundation of expressing numbers that cannot be represented on the real number line. The crucial property of the imaginary unit is that when squared, it equals negative one: \(i^2 = -1\).
This definition allows mathematicians and scientists to work with and quantify the square roots of negative numbers, which are otherwise undefined in the real number system. The imaginary unit \(i\) is especially useful in fields like engineering, physics, and complex analysis.
By incorporating \(i\) into mathematical expressions, we can expand our number system to include complex numbers, which have both real and imaginary parts. Complex numbers can be written in the form \(a+bi\), where \(a\) is the real component and \(b\) is the imaginary component.
This definition allows mathematicians and scientists to work with and quantify the square roots of negative numbers, which are otherwise undefined in the real number system. The imaginary unit \(i\) is especially useful in fields like engineering, physics, and complex analysis.
By incorporating \(i\) into mathematical expressions, we can expand our number system to include complex numbers, which have both real and imaginary parts. Complex numbers can be written in the form \(a+bi\), where \(a\) is the real component and \(b\) is the imaginary component.
Simplifying Expressions
Simplifying mathematical expressions involving complex numbers often involves rewriting expressions with imaginary units in a cleaner or more conventional form.
When dealing with square roots of negative numbers, as seen in our example, it's important to convert these square roots to expressions involving the imaginary unit \(i\).
For example, the square root of \(-25\), indicates that you should express it as \(\sqrt{-25} = 5i\). This use of \(i\) allows us to handle complex expressions easily.
After these conversions, simplifying the entire expression usually involves basic algebraic operations like multiplication, division, and combining like terms.
When dealing with square roots of negative numbers, as seen in our example, it's important to convert these square roots to expressions involving the imaginary unit \(i\).
For example, the square root of \(-25\), indicates that you should express it as \(\sqrt{-25} = 5i\). This use of \(i\) allows us to handle complex expressions easily.
After these conversions, simplifying the entire expression usually involves basic algebraic operations like multiplication, division, and combining like terms.
- Substitute where necessary using known simplifications: \(\sqrt{-16} = 4i\) and \(\sqrt{-81} = 9i\).
- Replace all components in the original expression with their simplified forms.
- Perform arithmetic operations like multiplications or divisions with care, especially keeping track of \(i^2\) because \(i^2 = -1\).
Square Roots of Negative Numbers
Square roots of negative numbers are not defined among real numbers due to the fact that squaring any real number (whether positive or negative) results in a positive number.
However, with the creation of the imaginary unit \(i\), we can represent these square roots. For instance, the square root of \(-1\) becomes \(i\), allowing us to expand our mathematical operations.
However, with the creation of the imaginary unit \(i\), we can represent these square roots. For instance, the square root of \(-1\) becomes \(i\), allowing us to expand our mathematical operations.
Understanding Square Roots of Negative Numbers:
- The square root of any negative number \(-x\) (where \(x > 0\)), can be expressed as \(i\sqrt{x}\).
- This gives rise to expressions like \(\sqrt{-25} = 5i\), where we convert \(\sqrt{-25}\) to \(5i\), utilizing \(i\) to handle the negative.
- This approach makes it possible to work within a complex number framework, handling expressions that involve such roots seamlessly.
Other exercises in this chapter
Problem 33
Exer. 11-46: Simplify. $$ \left(4 a^{3 / 2}\right)\left(2 a^{1 / 2}\right) $$
View solution Problem 33
Exer. 33-40: Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true for all real numbers \(a, b\), \(c\), and \(d\),
View solution Problem 34
Express as a polynomial. $$ (\sqrt{x}+\sqrt{y})^{2}(\sqrt{x}-\sqrt{y})^{2} $$
View solution Problem 34
Exer. 11-46: Simplify. $$ \left(-6 x^{7 / 5}\right)\left(2 x^{8 / 5}\right) $$
View solution