Problem 21 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 100^{-\frac{3}{2}} $$

Step-by-Step Solution

Answer

The simplest form of \(100^{-\frac{3}{2}}\) is \(\frac{1}{1000}\).

Step 1: Understand the expression

The expression given is \(100^{-\frac{3}{2}}\). This is an expression involving a negative exponent and a fractional exponent, which we need to simplify.

Step 2: Convert the negative exponent

A negative exponent indicates the reciprocal of the base raised to the positive of that exponent. Therefore, \(100^{-\frac{3}{2}}\) becomes \(\frac{1}{100^{\frac{3}{2}}}\).

Step 3: Simplify the fractional exponent

Next, simplify the expression with the fraction exponent \(100^{\frac{3}{2}}\). The exponent \(\frac{3}{2}\) means raising the base to the 1/2 power (square root) and then cubing the result.

Step 4: Find the square root

Calculate the square root of 100. Since \(\sqrt{100} = 10\), the expression becomes \(10^3\).

Step 5: Cube the result

Calculate \(10^3\). Since \(10^3 = 1000\), this means \(100^{\frac{3}{2}} = 1000\). We can now write the expression as \(\frac{1}{1000}\).

Step 6: Present the answer

The simplified form of the expression \(100^{-\frac{3}{2}}\) is \(\frac{1}{1000}\).

Key Concepts

Understanding Negative Exponents

Negative exponents are a fascinating concept that turn the base into its reciprocal. When you see a negative exponent, like in the expression \(100^{-\frac{3}{2}}\), this means the base, 100, is to be represented as the reciprocal of its positive exponent version. In simpler terms:

• Flip the base. The negative sign tells you to take \(\frac{1}{\text{base}}\).
• Change the exponent to positive. So, \(100^{-\frac{3}{2}} = \frac{1}{100^{\frac{3}{2}}}\).

This reciprocal transformation prepares the base for further calculations, especially useful when simplifying expressions. Working with negative exponents can initially seem puzzling, but remembering the "reciprocal" rule helps make these calculations straightforward.

Comprehending Fractional Exponents

Fractional exponents come into play as a more elegant way to express operations like roots. They essentially mean taking a root and then powering the result or vice versa. For instance, in \(100^{\frac{3}{2}}\):

• The fraction \(\frac{3}{2}\) indicates you're dealing with both a square root and a cube operation.
• The bottom number, 2, is the root. It tells you to take the square root of the base, 100.
• The top number, 3, is the exponent. After finding the square root, you need to cube the result.

Thus, this expression simplifies as follows: Start by calculating the square root of 100, which is 10. Then, cube 10 to get 1000. Therefore, \(100^{\frac{3}{2}} = 1000\). Fractional exponents can initially seem complex, but they are just another way to describe multiple sequential operations.

The Art of Simplifying Expressions

Simplifying expressions is about taking a complex problem and breaking it down into more manageable parts. For expressions with rational exponents, follow systematic steps:

• Handle each component separately, starting from negative exponents. Apply the reciprocal to make the exponent positive.
• Next, address the fractional exponent by performing the indicated root and powers.
• Finally, simplify the results to the most compact form.

For instance, in simplifying \(100^{-\frac{3}{2}}\); after converting the negative exponent, simplify the fractional exponent as previously explained. The resulting expression is \(\frac{1}{1000}\), a straightforward rational number. By breaking down tasks into smaller steps, simplifying even the most intimidating expressions becomes a less daunting process.