Problem 27 - Algebra for College Students

Question

Evaluate each numerical expression. \(-25^{\frac{3}{2}}\)

Step-by-Step Solution

Answer

The expression evaluates to -125i in the complex number system.

Step 1: Understand the Expression

The expression -25^{\frac{3}{2}} involves a base number of -25 raised to the power of \(\frac{3}{2}\). This means we need to find the square root of -25 and then raise the result to the third power.

Step 2: Check for Real Solutions

Since the expression involves \(-25\) and we need to take its square root (denoted by the denominator 2 in the exponent), we should consider that the square root of a negative number is not defined in the real number system. This problem implies imaginary or complex numbers.

Step 3: Simplify using Complex Numbers

To solve this problem within the complex number system, we express -25 as \(-1 \times 25\). The square root of each part would be \(\sqrt{-1} \times \sqrt{25}\). This results in i \times 5. Hence, the square root of -25 is 5i.

Step 4: Raise to the Third Power

Now we take our result from Step 3 and raise it to the power of 3: \((5i)^3\). First, calculate \(5^3 = 125\), and we have \((i)^3 = -i\). So, \((5i)^3 = 125i^3 = 125(-i) = -125i.\)

Key Concepts

Imaginary Numbers

Imaginary numbers are numbers that when squared yield a negative result. This is contrary to all positive and zero-valued numbers which, when squared, yield a non-negative result. Imaginary numbers are founded upon the imaginary unit, represented by the symbol \(i\).

• The imaginary unit \(i\) is defined such that \(i^2 = -1\).
• Using \(i\), we can define imaginary numbers like \(5i, -2i,\) and so on.

When you encounter the square root of a negative number, the solution lies in using imaginary numbers. For instance, the square root of \(-25\) can be expressed as \(5i\) because \(\sqrt{-25} = \sqrt{-1} \times \sqrt{25} = 5i\). Imaginary numbers become essential in fields such as engineering and physics, offering solutions to problems that would otherwise have none.

Exponentiation

Exponentiation refers to the process of raising a number (the base) to the power of another number (the exponent). This is expressed in the form \(a^b\), which means multiply \(a\) by itself \(b\) times. In the expression \(-25^{\frac{3}{2}}\), exponentiation involves not only multiplying but also considering fractional powers meaning roots.

• A fractional exponent like \(\frac{3}{2}\) means we first take a square root (denominator 2) and then cube the result (numerator 3).
• Since exponentiation in this context involves imaginary numbers, calculating \((-25)^{\frac{1}{2}} = 5i\) is the first part.
• The next step is raising this result \(5i\) to the 3rd power.

This process shows how exponentiation can be applied even with negative and imaginary numbers.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For positive numbers like 25, the square root is straightforward, giving us \(\sqrt{25} = 5\). But for negative numbers, we need to delve into the realm of imaginary numbers.

• The square root of a negative number involves the imaginary unit \(i\), where \(\sqrt{-1} = i\).
• For example, \(\sqrt{-25} = \sqrt{-1 \times 25} = i \times \sqrt{25} = 5i\).

Square roots are a critical component of complex arithmetic, allowing us to solve quadratic equations and explore complex numbers.

Algebraic Expressions

Algebraic expressions combine numbers and variables with operations like addition, subtraction, multiplication, division, and exponentiation. In our problem with \(-25^{\frac{3}{2}}\), we interpret the expression to involve both exponentiation and the imaginary unit.

• Here, \(-25\) serves as the base in the algebraic expression, with \(\frac{3}{2}\) as its exponent.
• Through an understanding of complex numbers, we manage the algebraic expression to accommodate imaginary solutions.

Such conventions allow the solving of expressions involving complex numbers, broadening the scope of possible solutions and showcasing the importance of arithmetic rules in extending mathematical understanding.