Problem 85 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(z^{4}\right)^{10} $$

Step-by-Step Solution

Answer

\( z^{40} \)

Step 1: Understand the Power of a Power Rule

When raising an exponent to another exponent, you multiply the exponents. The rule can be expressed as \( (a^m)^n = a^{m imes n} \).

Step 2: Apply the Rule to the Given Expression

Using the power of a power rule, apply it to the expression \( \left(z^4\right)^{10} \). This means multiplying the exponents in the expression: \( 4 \times 10 \).

Step 3: Perform the Multiplication

Calculate \( 4 \times 10 \), which equals 40.

Step 4: Write the Simplified Expression

Substitute the multiplied exponent back into the expression. Thus, \( \left(z^4\right)^{10} \) simplifies to \( z^{40} \).

Key Concepts

Power of a Power Rule

The power of a power rule is a foundational concept in algebra that helps simplify expressions where an exponent is raised to another exponent. When dealing with such expressions, you simply multiply the exponents to form a single power. This is easily remembered by the formula \( (a^m)^n = a^{m \times n} \).

This rule is particularly useful when dealing with repeated multiplication. For instance, consider the expression \( \left(z^4\right)^{10} \). To simplify it, you multiply the inner exponent 4 by the outer exponent 10. So the expression becomes \( z^{4 \times 10} \). By using this rule, you can simplify complex expressions very efficiently.

Simplifying Expressions

Simplifying expressions means making them as concise as possible while keeping them mathematically equivalent to the original. This often involves reducing the expression to its simplest form or combining like terms.

For algebraic expressions involving exponents, simplifying often uses the exponent rules, such as the power of a power rule. In our example, converting \( \left(z^4\right)^{10} \) to \( z^{40} \) simplifies the expression by using fewer steps and less space. This process not only makes the expression easier to work with but also reduces the potential for error in further calculations.

Multiplying Exponents

Multiplying exponents involves applying the power of a power rule to compress multiple layers of exponents into a single layer. When you see an expression like \((a^m)^n\), multiply the exponents, \(m\) and \(n\), to simplify it into \(a^{m \times n}\).

In practical scenarios, consider \( \left(z^4\right)^{10} \). You multiply 4 by 10 resulting in \( z^{40} \). This technique doesn't change the value of the expression but simplifies the computation significantly.

• First, identify each exponent.
• Next, apply the multiplication of the exponents.
• Finally, replace with the new simplified exponent.

Once mastered, multiplying exponents becomes an automatic and straightforward step in simplifying expressions.