Problem 10 - A Graphical Approach to College Algebra

Question

Simplify each expression. Leave answers with exponents. $$\left(-4 m^{3} n^{9}\right)^{2}$$

Step-by-Step Solution

Answer

16m^6n^{18}

Step 1: Apply the Power of a Power Rule

When an expression with exponents is raised to another power, use the rule \((a^m)^n = a^{m \cdot n}\). In this case, apply the rule to each factor inside the parentheses: \[ (-4 m^3 n^9)^2 = (-4)^2 (m^3)^2 (n^9)^2 \]

Step 2: Simplify Each Factor

Now simplify each part of the expression from Step 1:- For \((-4)^2\), calculate the power: \((-4)^2 = 16\).- For \((m^3)^2\), multiply the exponents: \(m^{3 \cdot 2} = m^6\).- For \((n^9)^2\), multiply the exponents: \(n^{9 \cdot 2} = n^{18}\).

Step 3: Combine the Simplified Parts

Combine all the simplified parts together to get the final expression:\[ 16 m^6 n^{18} \] This is the simplified form of the original expression.

Key Concepts

Power of a Power Rule

The "Power of a Power Rule" is a fantastic tool for dealing with exponents. Imagine having a mathematical expression where you not only have a base raised to a power but that whole expression is again raised to another power. It might look complicated at a glance, but this rule comes to the rescue.

Here's how it works: If you have something like \((a^m)^n\), you can simply multiply the exponents together to get \(a^{m \cdot n}\). This is because the operation essentially repeats the multiplication of the base by itself multiple times, equivalent to the product of the exponents.

• Example: \((m^3)^2 = m^{3 \cdot 2} = m^6\)
• This means you'll multiply the number of times the base is used as a factor.

Understanding this rule truly streamlines the process of simplifying expressions and saves tons of time. It’s really neat!

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to a simpler or more concise form without changing their value. This is where we put those rules of exponents to good use!

By applying the rules, such as the "Power of a Power Rule," we handle each part of the expression step by step. For instance, when you encounter \((-4 m^3 n^9)^2\), you take each component:

• For \((-4)^2\), compute the power, which results in 16 because multiplying negative numbers gives a positive result when the exponent is even.
• For \((m^3)^2\), multiply the exponents to get \(m^6\).
• Similarly, for \((n^9)^2\), multiply the exponents to get \(n^{18}\).

By breaking it down, you end up with 16 \(m^6 n^{18}\), where each factor is simplified, but none of the original values are altered.

Algebraic Expressions

Algebraic expressions are like puzzles waiting to be solved! These expressions include numbers, variables, and operation symbols. They can be as simple as \(2x + 3\) or more complex like \((-4 m^3 n^9)^2\). Understanding how to manipulate these expressions is key to solving algebra problems effortlessly.

Algebraic expressions can be simplified using several rules:

• Combine like terms - terms that have the same variable raised to the same power.
• Use exponent rules - these help simplify expressions with powers.
• Factor or distribute - helps in both breaking down and building up expressions.

By practicing these techniques, you'll become more comfortable with breaking down and reconstructing expressions. You will also see the beauty in how algebra can express complex ideas simply. It’s amazing how a jumble of numbers and letters can form meaningful solutions to real-world problems.