Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression. $$\left(\frac{x}{8}\right)^{-3}$$

Step-by-Step Solution

Answer

The simplified form of \( \left( \frac{x}{8} \right) ^{-3} \) is: \( \frac{512}{x^3} \).

Step 1: Understand Exponent of -1

When any number or term is raised to the power -1, it means the reciprocal of that number. \(\left( \frac{x}{8} \right) ^{-1} = \frac{8}{x} \).

Step 2: Understand Power Law

\((a^n)^m = a^{n \cdot m}\) which indicates that when we are raising a power to another power, we multiply the exponents which also applies to negative exponents. To simplify \(\left (\frac{x}{8} \right )^{-3} \) we will take the cube of \(\frac{8}{x} \) which means \(\left( \frac{8}{x} \right) ^3\).

Step 3: Simplify

Let's compute \(\left( \frac{8}{x} \right) ^3\). It is the same as multiplying the numerator and the denominator by themselves three times. This leads to \( \frac{8^3}{x^3} = \frac{512}{x^3}\).

Key Concepts

Understanding Reciprocals

When we talk about reciprocals, we're discussing numbers that, when multiplied together, yield the number 1. Essentially, the reciprocal of a fraction is achieved by flipping the numerator and the denominator.

• For instance, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
• Similarly, the reciprocal of a whole number \( c \) is \( \frac{1}{c} \).

Reciprocals become particularly useful when handling expressions with negative exponents.
When you see a negative exponent, like \(-1\), it's a signal to find the reciprocal of that base. For example, \( \left( \frac{x}{8} \right)^{-1} \) results in \( \frac{8}{x} \), because you're flipping \( \frac{x}{8} \).
This simple switch makes it easier to deal with various calculations, especially when simplifying expressions or fractions with exponents.

Exploring the Power Law Concept

The power law in mathematics lets us figure out the result when raising a power to another power. This rule is crucial because it simplifies expressions with exponents.

• The general form can be described as \((a^n)^m = a^{n\cdot m}\).

This formula advises us to multiply the exponents together.
For example, if we have \((a^2)^3\), we compute it as \(a^{2\cdot 3} = a^6\). The same principle can apply to negative exponents.
In our exercise \(\left( \frac{x}{8} \right)^{-3}\), we treat the expression as a power to another power. Using the power law helps us convert it to \(\left( \frac{8}{x} \right)^3\).
This simplifies because we already know \(\left( \frac{x}{8} \right)^{-1}\) leads to \(\frac{8}{x}\), and raising that to the third power is straightforward.

Simplifying Expressions with Exponents

Simplifying expressions with exponents can initially seem challenging, yet it's manageable with the right approach. The goal is to reduce the expression to its simplest form by applying rules for exponents and reciprocals.

• Take each factor and simplify it using known rules, such as reciprocals and the power law.
• Always break down complex expressions gradually, ensuring clarity at each step.

In the original exercise, simplifying \( \left( \frac{8}{x} \right)^3 \) means multiplying both the numerator and the denominator by themselves three times.
This turns into \( \frac{8 \times 8 \times 8}{x \times x \times x} \), giving us \( \frac{8^3}{x^3} \).
Hence, by computing \(8^3 = 512\), we find our expression simplifies to \( \frac{512}{x^3} \).
Remember to perform multiplication step by step, ensuring accurate results while reinforcing your understanding of exponent rules.