Problem 56 - College Algebra

Question

Simplify each exponential expression $$ \left(10 x^{2}\right)^{-3} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\left(10 x^{2}\right)^{-3}\) is \(1/(1000x^6)\).

Step 1: Applying the Power of a Power Property

Use the power of a power property that states \((a^m)^n = a^{mn}\) to get: \(10^{-3} * (x^2)^{-3}\).

Step 2: Simplify the Numerical Fraction

To simplify \(10^{-3}\), we take the reciprocal of \(10^3\) to get \(1/10^3 = 1/1000\).

Step 3: Simplify the Variable Fraction

Then simplify \((x^2)^{-3}\) to \(x^{-6}\), since the product of the exponents is \(-6\). If the exponent is negative, take the reciprocal. Thus, \(x^{-6} = 1/x^6\).

Step 4: Combine the Results

Combine all the results from above steps to express the simplified form of the given expression. It becomes \(1/1000 * 1/x^6\).

Key Concepts

Power of a Power Property

The power of a power property is a handy tool when dealing with exponents. It states that when you take a power of a power, you multiply the exponents. In mathematical terms, it's written as \((a^m)^n = a^{mn}\). Applying this rule helps to simplify expressions where exponents are involved.
For example, if you have \((a^2)^3\), it becomes \(a^{2*3}\) or simplified to \(a^6\). This property is essential for breaking down expressions, making them easier to work with and understand.
By using this property correctly, you can systematically manage complex exponential expressions, avoiding confusion and errors.

Negative Exponents

Negative exponents might look intimidating at first, but they are quite straightforward. A negative exponent essentially means "take the reciprocal". In mathematical language, \(a^{-n}\) is the same as \(1/a^n\).
This is why when you see something like \(10^{-3}\), it becomes \(1/10^3\) or \(1/1000\). Similarly, if you encounter a variable with a negative exponent, say \(x^{-2}\), it translates to \(1/x^2\).
Understanding negative exponents is integral in simplifying fractions and expressions, making complex problems more manageable.

Simplifying Fractions

Simplifying fractions involves breaking down complex components into simpler parts. When you have exponents in a fraction, it often involves using the property of negative exponents.
Take for example the expression \(10^{-3}\); by simplifying using the reciprocal, it becomes \(1/10^3 = 1/1000\). Similarly, a variable like \(x^{-6}\) becomes \(1/x^6\).
Once simplified, these parts can be easily combined to form a singular, straightforward fraction. This process is crucial for solving and simplifying algebraic expressions efficiently.

Algebraic Simplification

Algebraic simplification is all about making complex expressions easier to handle. Using laws of exponents and properties of fractions, we aim to express equations in their simplest form.
For example, the expression \((10 x^2)^{-3}\) initially looks intricate. Yet, by applying the power of a power property, negative exponents, and fraction simplification, it reduces to a simpler form \(1/1000 * 1/x^6 \). This becomes \(1/(1000x^6)\).
Through step-by-step simplification, complex algebraic expressions can be transformed into clearer, more understandable results.