Problem 39 - Algebra 1

Question

Rewrite the expression with positive exponents. $$(-4 x)^{-3}$$

Step-by-Step Solution

Answer

The expression \(-4x^{-3}\) with positive exponents is \(\frac{1}{-64x^3}\).

Step 1: Identify the Base and Power

In the given expression, \((-4x)\) is the base and -3 is the power.

Step 2: Apply Negative Exponent Rule

According to the rule \(a^{-n} = \frac{1}{a^n}\), the expression can be rewritten as \(\frac{1}{(-4x)^3}\)

Step 3: Evaluate Power

Calculate the cube of \(-4x\), which gives us -64\(x^3\). Replace this value in the expression.

Step 4: Final Expression

The final expression after replacing the calculated value is \(\frac{1}{-64x^3}\).

Key Concepts

Understanding Negative Exponents

Exponents tell us how many times to multiply a number by itself.Negative exponents indicate that we should use the reciprocal of the base raised to the absolute value of the exponent.When you see a negative exponent, remember the following rule:

• For any non-zero number 'a' and a positive integer 'n', \(a^{-n} = \frac{1}{a^n}\).

This means that instead of multiplying 'a' by itself 'n' times, with a negative exponent, you're actually dividing by 'a' 'n' times.
So, for example,

• \((2)^{-3} = \frac{1}{2^3} = \frac{1}{8}\).
• This rule helps switch a "negative" repetition into a "division" process.

In our original exercise, \((-4x)^{-3}\), understanding negative exponents means expressing it in reciprocal form as \(\frac{1}{(-4x)^3}\) before proceeding to any further calculations.

Exploring Power of a Product

When you have an expression like \((-4x)^{-3}\), you're dealing with both a negative exponent and a product inside the base.The "power of a product" rule helps manage such expressions. It states:

• If you have a product \(ab\) raised to a power \(n\), you can apply the exponent to both the 'a' and 'b' separately: \((ab)^n = a^n \cdot b^n\).

This means when simplifying \((-4x)^{-3}\), you can break it down as:

• \((-4)^3\) and \(x^3\), and both parts are affected by the exponent.

By doing this, you simplify the expression as \(\frac{1}{(-4)^3 \cdot x^3}\), revealing each part clearly.
This step is crucial in making complex expressions more manageable, allowing you to calculate final values easily.

Mastering Simplifying Expressions

Simplifying expressions is about breaking down complex algebraic terms into simpler forms while following algebraic rules.This process includes understanding exponents, distributing them across variables and constants, and rewriting the expression in the simplest way possible.
In the original exercise, we've already rewritten \((-4x)^{-3}\) as \(\frac{1}{(-4)^3 \cdot x^3}\). Now it's time to evaluate it:

• Calculate \((-4)^3\) which is \(-64\).
• This results from multiplying -4 by itself three times: \(-4 \cdot -4 \cdot -4 = -64\).
• Combine it with \(x^3\), forming \(-64x^3\).

So, simplify the original expression to \(\frac{1}{-64x^3}\).Each step builds upon the previous one to ensure clarity and accuracy, helping you achieve the simplest form of any expression effectively.