Problem 43 - Algebra for College Students

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-4} \sqrt{-16}$$

Step-by-Step Solution

Answer

The simplified expression is \( -8 \).

Step 1: Express each square root in terms of i

Rewrite the given expression \( \sqrt{-4} \sqrt{-16} \) by expressing the square roots as imaginary numbers. This means using \( i = \sqrt{-1} \). \[ \sqrt{-4} = i\sqrt{4} \, \text{and} \, \sqrt{-16} = i\sqrt{16} \]

Step 2: Multiply the expressions

Multiply the expressions you got from Step 1: \[ (i\sqrt{4})(i\sqrt{16}) = i^2\sqrt{4\times16} \]

Step 3: Calculate the square root and simplify

Calculate the product of the square roots and simplify: \[ i^2\sqrt{64} = (-1)\times8 = -8 \]

Step 4: Conclusion

Combine all the above steps, and the solution to \( \sqrt{-4} \sqrt{-16} \) is \( -8 \).

Key Concepts

Square Roots

The concept of square roots is foundational, especially when dealing with imaginary numbers. A square root of a number is a value that, when multiplied by itself, gives the original number. For any positive number, you can find two square roots, one positive and one negative. For example, the square roots of 4 are 2 and -2 because both squared result in 4.
When it comes to negative numbers, the square root concept introduces imaginary numbers. For instance, the square root of -4 isn't a real number since no real number squared gives -4. Thus, we use the imaginary unit, represented as \(i\), such that \(i^2 = -1\). Hence, \(\sqrt{-4} = i\sqrt{4} = 2i\).
Square roots extend to complex operations and are key in solving equations involving negative values under roots, requiring an understanding of both real and imaginary components. This extends to working with the radical sign \(\sqrt{}\), and the presence of negative values under this sign indicates a move into the realm of imaginary numbers.

Complex Numbers

Complex numbers are a blend of real numbers and imaginary numbers. Represented as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part, complex numbers expand our number system beyond the real line. The imaginary unit \(i\) is crucial because it lets us work with the square roots of negative numbers.
In problems like \(\sqrt{-4} \sqrt{-16}\), expressing each square root in terms of \(i\) is essential. For \(\sqrt{-4}\), you express it as \(i\sqrt{4}\), and similarly, \(\sqrt{-16}\) becomes \(i\sqrt{16}\). These expressions rely on the definition of the imaginary unit \(i\).
Understanding complex numbers means fully grasping both components. They enable complex equations to have solutions where none exist using only real numbers. They also have practical applications in fields like engineering and physics, where waveforms and alternating current calculations frequently involve complex numbers.

Multiplying Radicals

Multiplying radicals, especially with imaginary numbers, follows specific algebraic rules. A radical is the root of a number, and when multiplying them, it's important to manage both the numbers under the roots and any coefficients outside.
Taking a look at \(\sqrt{-4} \sqrt{-16}\), you've expressed each radical individually with imaginary units first. This gives us: \((i\sqrt{4})(i\sqrt{16})\). Multiplying these expressions involves multiplying the coefficients and the radicals themselves.
The process involves:

• Calculating \(i^2\), which equals -1, due to the definition of the imaginary unit.
• Taking the product of the square roots \(\sqrt{4 \times 16} = \sqrt{64} = 8\).
• Combining these results: \(-1 \times 8 = -8\).

Care must be taken in handling the imaginary unit correctly and ensuring proper calculations involving the radicals. This process helps in simplifying expressions, merging both real and imaginary elements smoothly in complex number multiplication.