Problem 125 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \left(m^{4}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

The expression \((m^4)^{1/2}\) simplifies to \(m^2\).

Step 1: Understand the Problem

This exercise requires simplifying the expression \((m^4)^{1/2}\), where \(m\) represents a positive real number.

Step 2: Apply the Power of a Power Rule

The power of a power rule in exponents states that \((a^m)^n = a^{m \cdot n}\). In this case, apply the rule to the expression \((m^4)^{1/2}\), resulting in the expression \(m^{4 \cdot \frac{1}{2}}\).

Step 3: Simplify the Exponent

Calculate the exponent by multiplying 4 with \(\frac{1}{2}\). This gives us \(4 \cdot \frac{1}{2} = 2\). So, the expression simplifies to \(m^2\).

Key Concepts

Power of a Power Rule

The Power of a Power Rule is a fundamental principle in exponentiation that helps simplify expressions where exponents are involved. If you have an expression like

• \((a^m)^n \)

This rule lets us multiply the exponents together, transforming it into:

• \(a^{m \cdot n} \)

This works because multiplying an exponent by itself multiple times is like repeated multiplication, so multiplying the exponents together gives you the total power.
Using this rule, complex expressions with nested exponents can be simplified, making it a powerful tool in algebra. For instance, in the case of

• \((m^4)^{1/2} \)

you apply the Power of a Power Rule to get:

• \(m^{4 \cdot \frac{1}{2}} \)

This simplification results in a more straightforward form to work with.

Simplifying Expressions

Simplifying expressions is a process to make mathematical expressions easier to work with and understand. For algebraic expressions, this often involves following certain rules known as properties of exponents. These include:

• The Power of a Product Rule:
  • \((ab)^m = a^m b^m\)
• The Power of a Power Rule:
  • \((a^m)^n = a^{m \cdot n}\)

In the expression

• \((m^4)^{1/2} \)

applying these rules helps break complex expressions into simpler forms. As part of the step-by-step solution process, using these exponent rules can strategically reduce the complexity of an equation or expression, resulting in something much more manageable, like

• \(m^2 \)

Simplifying typically involves reducing terms and coefficients, consolidating like terms, and performing arithmetic operations on exponents. This makes it easier to see and understand the fundamental structure and outcomes of an expression.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations (like addition and multiplication) without an equals sign. They are used to represent real-world scenarios and mathematical relations. A simple algebraic expression can look like

• \(3x + 2 \)

where

• \(3x \)

indicates 3 multiplied by the variable \(x\), and 2 is a constant term.
Working with algebraic expressions often involves simplifying or rearranging terms. Understanding expressions is key to solving equations, as they represent one side of an equation balancing out the other.
In expressions like

• \((m^4)^{1/2} \)

we focus by simplifying it using previously discussed exponent rules. In algebra, it's essential to learn how to structure, identify, and transform these representations, enabling us to evaluate scenarios and solve for unknown variables efficiently.