Problem 33 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. \(\left[(-4)^{5}\right]^{3}\)

Step-by-Step Solution

Answer

The simplified expression is \((-4)^{15}\)

Step 1: Identify the base and the exponents

In the given expression \(\left[(-4)^{5}\right]^{3}\), the base is -4, and it's raised to the power 5, which in turn is raised to the power 3.

Step 2: Apply the Power of a Power Property

Using the property of exponents that states \((a^m)^n = a^{m \cdot n}\), apply this to the given expression. Multiply the exponents 5 and 3 together to get a new exponent: \((-4)^{5 \cdot 3}\)

Step 3: Simplify to a single power

Multiply the exponents: \((-4)^{15}\)

Key Concepts

Power of a Power

When dealing with exponents, sometimes you'll find an exponent raised to another exponent. This is known as the "power of a power" rule. It helps simplify expressions like \((a^m)^n\). Here's what you do: multiply the exponents together, while keeping the base the same.
For example, in the expression \(((-4)^5)^3\), you see a base of \(-4\) raised first to the exponent 5, then to another exponent 3. Using the rule, you'll multiply 5 by 3 to get the new exponent, resulting in \((-4)^{15}\). This rule makes it easier to work with complex exponent expressions, simplifying them into a single power.

• Use this rule whenever you have nested exponents.
• Remember: Multiply, don’t add exponents in power of a power.

Base and Exponent

Understanding the terms 'base' and 'exponent' is crucial for mastering exponent rules. The base is the number being multiplied, while the exponent tells you how many times to multiply the base by itself.
Take \((-4)^5\) for example: \(-4\) is the base, and 5 is the exponent. This means you multiply \(-4\) by itself a total of 5 times. It's like having several layers of repeated multiplication.

• The base can be any real number, even negative, like \(-4\) in our exercise.
• The exponent is usually a positive integer, telling how many times to use the base in the multiplication.

Knowing these terms makes it easier to follow exponent rules and simplify expressions properly.

Simplifying Exponents

Simplifying exponents means making an expression easier to work with, usually by writing it as a single power. After applying rules like "power of a power," you often end up with expressions that can be simplified further.
In our example, simplifying \((-4)^{5}\) raised to the power of 3 gave us \((-4)^{15}\). This single exponent expression is simpler and more practical when solving equations or performing calculations.

• Always aim to reduce expressions to the simplest form.
• Simplifying helps in quicker calculations and reduces errors.

Mastering the art of simplification is essential for working efficiently with exponential expressions in algebra and beyond.