Problem 15 - Algebra 1

Question

Use the power of a power property to simplify the expression. $$ \left(m^{4}\right)^{8} $$

Step-by-Step Solution

Answer

So, the simplified form of \( \left(m^{4}\right)^{8} \) is \( m^{32} \).

Step 1: Apply the Power of a Power property

In order to simplify \( \left(m^{4}\right)^{8} \), apply the power of a power property which is \( \left(a^{m}\right)^{n} = a^{m \times n}\). Here, the base is \(m\), \(m^8\) is the inner exponent, and 4 is the outer exponent. Multiply the exponents together.

Step 2: Perform the Calculation

Perform the multiplication of the exponents. \(4 \times 8 = 32\). Therefore, the simplified version of \( \left(m^{4}\right)^{8} \) is \( m^{32} \).

Key Concepts

Understanding the Power of a Power Property

When working with exponents, the power of a power property is a handy tool. This property helps you simplify expressions where an exponent is raised to another power.
For example, consider the expression \((m^{4})^{8}\). Here, you have a base \(m\) with an inner exponent of 4, further raised to an outer exponent of 8. To simplify, use the power of a power property:

• \((a^{m})^{n} = a^{m \times n}\)

Applying this, simply multiply the exponents: \(4 \times 8\), resulting in \(m^{32}\).
This simplifies your expression to one neat exponent instead of a nested one.

Simplifying Expressions

Simplifying expressions is about finding the most reduced form of a mathematical statement. It involves applying rules, such as the power of a power property, to make an expression easier to understand and work with.
For the expression \((m^{4})^{8}\), start by identifying the base and the exponents.

• The base is \(m\).
• The inner exponent is 4, and the outer exponent is 8.

By multiplying these exponents, you reduce the expression to a single exponential form: \(m^{32}\).
Simplifying in this way makes calculations more efficient and keeps expressions compact.

Understanding Exponent Multiplication

Exponent multiplication is crucial when simplifying expressions with powers. This process involves multiplying the numbers in the exponent portion of the expression,
which results from the power of a power property.

• In \((m^{4})^{8}\), you're dealing with two exponents: 4 and 8.
• Multiply them to find a combined exponent: \(4 \times 8 = 32.\)

The result \(m^{32}\) shows how multiplying the exponents simplifies the expression.
This technique helps tackle complex exponential expressions methodically, making them more manageable.