Problem 56 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$\left(10 x^{2}\right)^{-3}$$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is \(\frac{1}{1000x^6}\).

Step 1: Apply the negative exponent rule

The negative exponent rule states that \(a^{-n} = \frac{1}{a^n}\). By applying this rule, the expression \(\left(10 x^{2}\right)^{-3}\) can be rewritten as \(\frac{1}{\left(10 x^{2}\right)^{3}}\).

Step 2: Apply the power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). By applying this rule, the expression in the denominator, \(\left(10 x^{2}\right)^{3}\), can be rewritten as \(10^3 (x^2)^3\).

Step 3: Simplify the expression

Now, \(10^3 = 1000\) and \((x^2)^3 = x^6\) (using the power of power rule). Therefore, the denominator can be written as \(1000x^6\). Substituting this back into the fraction gives the final simplified form of the expression.

Key Concepts

Understanding the Negative Exponent Rule

The negative exponent rule is a fundamental concept in algebra that dictates how to handle expressions with exponents that are negative numbers. Essentially, when you encounter a term with a negative exponent, like \( a^{-n} \), the rule tells you to take the reciprocal of the term raised to the corresponding positive exponent. This means flipping the base term to the denominator and changing the exponent to positive, as in \( \frac{1}{a^n} \).

For example, applying this to \( 2^{-3} \) would yield \( \frac{1}{2^3} \) or \( \frac{1}{8} \). The rule simplifies the expression and removes any negative exponents, making calculation and further simplification easier. This rule is particularly useful when you are simplifying more complex expressions that involve multiple terms with negative exponents, as it provides a clear pathway to a positive solution.

Multiplying Through the Power of a Product Rule

When you have a product of terms raised to an exponent, the power of a product rule comes into play, guiding you on how to distribute the exponent to each term in the product. This rule states that for any terms a and b, and any exponent n, \( (ab)^n = a^n b^n \).

So if you have an expression like \( (3y)^4 \), you can apply this rule to simplify it to \( 3^4 y^4 \). This is very helpful as it allows each part of the product to be handled separately. The application of this rule is crucial for simplifying expressions and solving equations where terms are multiplied together before being raised to a power. Improper application can lead to mistakes, so remember each term within the parentheses gets the exponent.

Leveraging the Power of Power Rule

When dealing with exponential expressions, the 'power of a power' rule is vital for simplification. This rule states that when an exponent is raised to another exponent, you multiply the exponents together. Formally, for a base a and exponents m and n, \( (a^m)^n = a^{m \times n} \).

This simplification method makes working with multilayered exponential terms manageable. For instance, if you encounter \( (x^3)^2 \), you can simplify it to \( x^{3 \times 2} = x^6 \). This reduces the complexity of the expression and is essential for solving higher-level problems involving exponential growth, decay, or the laws of indices. It's important to not confuse this with other exponent rules, as each rule applies to different situations.