Problem 56 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Break down the expression

First, separate the expression \((10x^2)^{-3}\) into \(10^{-3}\) and \((x^2)^{-3}\). This uses the rule of exponential multiplication, which states that \(a^m*b^m = (a*b)^m\).

Step 2: Simplify terms with negative exponents

Next, simplify \(10^{-3}\) and \((x^2)^{-3}\). For \(10^{-3}\), it means the reciprocal of \(10^{3}\) or \(1/10^{3}\). For \((x^2)^{-3}\), again the rule of exponential multiplication is used to simplify it as \(x^{-6}\). A negative exponent means the reciprocal of the base, which is \(1/x^{6}\).

Step 3: Write the final simplified expression

The final result is \(1/10^{3} * 1/x^{6}\), combining the result of step 2. Simplify it to get \(1/(10^{3} * x^{6})\) or \(1/(1000x^{6})\) by calculating \(10^{3}\) as 1000.

Key Concepts

Negative Exponents

Negative exponents can seem a bit tricky at first, but they are pretty straightforward. When you see a negative exponent in an expression , it means that you have to take the reciprocal of the base. For example, if you have an expression like \( a^{-n} \), this is equivalent to \( \frac{1}{a^{n}} \). Think of the negative exponent as a way to "flip" the base into the denominator.

• \( 10^{-3} \) becomes \( \frac{1}{10^{3}} \)
• \( (x^2)^{-3} \) becomes \( \frac{1}{x^{6}} \) because \( x^{2} \) raised to \(-3\) power multiplies the exponents to give \( x^{(-3 \,\times\, 2)} \)

Understanding negative exponents is crucial when simplifying expressions, as it allows you to turn seemingly complex expressions into simpler ones by using the reciprocal concept.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This means getting rid of unnecessary parts and combining like terms.
Breaking down the expression \((10x^2)^{-3}\) means simplifying it step-by-step:

• First, break it into smaller parts using the rules of exponents: \( 10^{-3} \) and \( (x^2)^{-3} \).
• Then, simplify each part using negative exponent rules: \( \frac{1}{10^3} \) and \( \frac{1}{x^6} \).

In this exercise, simplifying helps us convert a complex-looking power expression into a more manageable fraction. The final simplified form of the expression becomes \( \frac{1}{1000x^6} \). It is like making everything nice and tidy in mathematical expression terms.

Exponential Multiplication

When it comes to exponential multiplication, it is important to understand that it involves applying multiplication rules to powers. This rule is foundational: the expression \( (a \cdot b)^m \) is equal to \( a^m \cdot b^m \).
This exercise used this concept:

• The expression \((10x^2)^{-3}\) was initially split into parts, \(10^{-3}\) and \((x^2)^{-3}\).
• Each part was then dealt with separately using the rules of multiplying exponents.

By breaking down the expression in this way, you multiplied the exponents correctly and ensured that each part of the expression was simplified properly before final combination.

Reciprocal of a Base

In mathematics, the reciprocal of a base is a way to "invert" numbers. Simply put, if you have a base \(b\), its reciprocal is \(\frac{1}{b}\). In algebra, you'll often see reciprocals when dealing with negative exponents.
For example, consider the reciprocal relationships used in this exercise:

• \( 10^{-3} \) is equivalent to \( \frac{1}{10^{3}} \), meaning the reciprocal of \( 10 \) raised to the third power.
• \( (x^2)^{-3} \) transforms to \( \frac{1}{x^6} \), which shows the reciprocal of \( x^6 \).

Grasping this concept helps you handle negative exponents and simplify expressions effectively. Reciprocal thinking makes the process of simplifying exponential expressions far more intuitive.