Problem 44 - Beginning and Intermediate Algebra

Question

Simplify. $$\left(-4 x^{-5} z^{-2}\right)^{-3}$$

Step-by-Step Solution

Answer

The simplified expression is \(-64 x^{15} z^6\).

Step 1: Apply the power rule to each term in the expression

First, we will apply the power rule to each term within the parentheses. Since the exponent is -3, we will multiply each term's exponent by -3. \((-4)^{-3} \cdot (x^{-5})^{-3} \cdot (z^{-2})^{-3}\)

Step 2: Simplify the terms

Now, we will simplify each term by multiplying its exponent by -3. \((-4)^{-3} = (-4)^3 = -64\) \((x^{-5})^{-3} = x^{(-5)(-3)} = x^{15}\) \((z^{-2})^{-3} = z^{(-2)(-3)} = z^6\)

Step 3: Combine the simplified terms

Finally, we will combine the simplified terms to get the simplified expression. \(-64 x^{15} z^6\) So, the simplified expression is \(-64 x^{15} z^6\).

Key Concepts

Power Rule

The power rule is a crucial concept in the world of exponents and powers. It allows us to simplify expressions by manipulating the exponents according to specific mathematical laws. When we have a term raised to an exponent, and this is again raised to another exponent, the power rule provides a quick way to simplify it.

The rule states:

• To find the new power, multiply the exponents. If you have \((a^m)^n\), it simplifies to \(a^{m imes n}\).

For instance, applying the power rule to \((-4)^m\), \((x^{-5})^n\), and \((z^{-2})^n\) enables us to perform multiplication of powers for each element in the expression separately.

When dealing with the power of a negative exponent, it is important to carefully multiply the exponents. For example, \((x^{-5})^{-3}\) becomes \(x^{15}\) since \(-5 imes -3 = 15\). This demonstrates the role of multiplication in transferring negative powers into positive, resulting in simplified terms.

Negative Exponents

Understanding negative exponents is paramount when working through expressions that involve them. A negative exponent indicates that instead of multiplying, you must take the reciprocal of the base. This rule turns confusing elements into comprehensible parts.

Here's how it works:

• A turning point for negative exponents is that \(a^{-n} = \frac{1}{a^n}\). A negative exponent flips or inverts the base into its reciprocal.

For example, consider \(-4\) raised to an exponent like \(-3\). This does not directly apply here because numbers themselves cannot be inverses. Instead, we can interpret this more complexly or break it down using the power rule.

When simplifying, these roles act together. If the negative exponents are nested within parentheses, as in our original problem, resolving the negative exponent involves switching the sign via multiplication, leading to positive results, such as turning \(x^{-5}\) into \(x^{15}\) under complex conditions with multiple steps.

Simplifying Expressions

Simplifying expressions is an essential skill in handling and solving mathematical problems. It involves reducing complex series of operations into a simpler and more digestible form. The process is about ensuring all components are easily interpretable for further application or conclusions.

The following principles help:

• Apply multiplication across exponents carefully. Take each term and resolve one step at a time.
• Turn all negative exponents into positive ones by using the power rule effectively.

Consider our initial expression: \(\left(-4 x^{-5} z^{-2}\right)^{-3}\). First, apply the rules of exponents separately.

For example, \((-4)^3 = -64\). Here, \(x^{-15}\) becomes \(x^{15}\) and \(z^6\) remains as it is once resolved.

By collecting these terms, the expression \(-64 x^{15} z^6\) results. This method collapses the equation into a much simplified, powerful expression that reflects the combined calculations performed at each stage.