Problem 49 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ \left(-w^{3}\right)\left(3 w^{2}\right)^{2} $$

Step-by-Step Solution

Answer

-9w^{7}

Step 1: Handle the brackets

In the given expression there are brackets, simplify the expression inside the brackets first: \((-w^{3})\) and \((3 w^{2})^{2}\). Resulting in \(-w^{3}\) and \(9w^{4}\) respectively.

Step 2: Multiply the results

Now multiply the results of Step 1, \(-w^{3} * 9w^{4}\), resulting in \(-9w^{7}\).

Step 3: Further Simplify the Result

The product obtained in Step 2, \(-9w^{7}\), is already in the simplest form and it cannot be simplified any further.

Key Concepts

Exponent Rules

Understanding the rules for exponents is like learning the grammar of algebra; it helps you handle algebraic expressions correctly and efficiently. One of the fundamental exponent rules is the 'Product of Powers' rule. It states that when you multiply two powers that have the same base, you can keep the base and add the exponents together. For example,\[ a^m \times a^n = a^{m+n} \]
This rule is especially handy when simplifying algebraic expressions involving multiple powers of the same variable. In the given exercise, we encounter the situation where a power is raised to another power. That's another exponent rule, known as the 'Power of a Power' rule, which tells us to multiply the exponents: \[ (a^m)^n = a^{m \times n} \]. When you multiply \( (-w^3) \times (3w^2)^2 \), you're applying this rule as well as the Product of Powers rule, leading to an exponent addition in the final step.
By understanding and applying these exponent rules, the process will not only be correct but also much more intuitive.

Multiplying Powers with Same Base

When you come across a multiplication involving powers with the same base, remember that you're not multiplying the bases themselves. Instead, you're combining the exponents. Here's a simple rule to guide you: when you multiply powers with the same base, add the exponents together to get a single power. This concept is pivotal in simplifying algebraic expressions, like the given exercise.
Take the example of mutiplying \( w^3 \) and \( w^4 \). According to the rule, \[ w^3 \times w^4 = w^{3+4} = w^7 \].
In the exercise, after applying the Power of a Power rule, you use this rule to combine \( -w^3 \) and \( 9w^4 \) into \( -9w^{3+4} \), which then simplifies to \( -9w^7 \). Undoubtedly, mastering this rule makes algebra a lot less daunting!

Simplifying Powers

Simplifying powers in algebra is a process of reducing expressions to their simplest form, often because it's easier to understand and work with. The goal is to minimize the number of terms and complexity of expressions without changing their value. It involves applying exponent rules such as the Product of Powers and Power of Powers. In the exercise, simplification was carried out step by step, starting with the elimination of brackets and then combining the powers.
For instance, after expanding \( (3w^2)^2 \), you simplify it to \( 9w^4 \). Then, you multiply it by \( -w^3 \) and simplify further by applying the rules for multiplying powers with the same base. The exercise shows that \( -w^3 \times 9w^4 \) simplifies to \( -9w^7 \), a single term exponent expression, which is as simple as it gets. Remember, simplifying powers is all about making equations cleaner and more manageable while sticking strictly to mathematical rules.