Problem 16 - Algebra 1

Question

Use the power of a product property to simplify the expression. $$ \left(2 m^{2}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(8 m^{6}\)

Step 1: Apply the Power of a Product Property

Using the power of a product property which states that \((ab)^n = a^n \cdot b^n \), the expression \((2 m^{2})^{3}\) can be rewritten as \(2^{3} \cdot (m^{2})^{3}\)

Step 2: Simplify the numerical part

Calculate the power of the number. Here \(2^{3} = 2 \cdot 2 \cdot 2 = 8\)

Step 3: Simplify the variable part

Simplify the variable part using the power of a power property which states that \((a^n)^m = a^{n \cdot m}\). Hence we can write \((m^{2})^{3} = m^{2 \cdot 3} = m^{6}\)

Step 4: Combine the simplified parts

Combine the simplified numerical part and variable part to get the final simplified expression. Hence, \(2^{3} \cdot (m^{2})^{3} = 8 m^{6}\)

Key Concepts

Simplifying Algebraic Expressions

When it comes to breaking down algebraic expressions into something more manageable, we call this process simplifying. It involves combining like terms, canceling common factors, and applying exponent rules. For instance, with an expression like \((2a^2b)^3\), simplifying means expanding and reducing the expression so it's easier to understand or use in further calculations. It's like decluttering a room, you want to organize and clear up any unnecessary complexity.

Exponent Rules

Exponent rules, also known as laws of exponents, are a set of guidelines that help us handle expressions involving exponents efficiently. There are several key rules: the product rule (\(a^m \cdot a^n = a^{m+n}\)), the quotient rule (\(a^m / a^n = a^{m-n}\)), the power of a power rule (\((a^m)^n = a^{m \cdot n}\)), and more. Understanding these rules allows for the proper manipulation of expressions to simplify or solve them. Always remember that the base number remains constant while you perform operations on the exponents.

Power of a Power Property

One of the more intriguing exponent rules is the power of a power property. This property tells us that when raising a power to another power, you multiply the exponents. For instance, \((x^a)^b = x^{a \cdot b}\). It's like stacking layers, where each new layer amplifies the previous. This is what makes exponential growth so rapid and significant. The power of a power rule comes in handy when simplifying expressions such as \((m^2)^3\), which simplifies to \(m^{2 \cdot 3} = m^6\).

Numerical Exponents

Dealing with numerical exponents is basically about understanding how many times to multiply a number by itself. The expression \(2^3\) tells us to multiply 2 by itself 3 times, leading to \(2 \times 2 \times 2 = 8\). What's interesting about numerical exponents is how they can drastically change the scale of a number. Small bases can grow to large numbers quickly as the exponent increases, illustrating the exponential 'power' in a very literal sense. Having a solid grasp of this simplifies larger problems and is essential when working with exponential growth or decay in various fields of study.