Problem 63 - Elementary and Intermediate Algebra

Question

Use the product and power rules for exponents to simplify each expression. $$ \left(t^{3}\right)^{4}\left(t^{2}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(t^{18}\).

Step 1: Apply the Power Rule

The power rule states that when you have a power to a power, you multiply the exponents. For \((t^3)^4\), the exponent is \(3\) and it's raised to the \(4\). So you multiply \(3 \times 4 = 12\), resulting in \(t^{12}\). Likewise, for \((t^2)^3\), the exponent is \(2\) and it's raised to \(3\). So, \(2 \times 3 = 6\), resulting in \(t^6\).

Step 2: Apply the Product Rule

The product rule states that when you multiply two powers with the same base, you add the exponents. Now, you have \(t^{12}\) and \(t^6\). Adding the exponents gives you \(12 + 6 = 18\), resulting in \(t^{18}\).

Key Concepts

Understanding the Power Rule

The power rule is essential for simplifying expressions with exponents. When you encounter a power raised to another power, you must multiply the exponents. This keeps the math tidy and straightforward. Let's consider the example \( (t^3)^4 \). Here, we first note the base, \( t \), and identify the two exponents involved: \( 3 \) and \( 4 \). By applying the power rule, calculate \( 3 \times 4 = 12 \). This simplification results in \( t^{12} \). Essentially, instead of getting entangled with multiple layers of exponents, this method allows you to reduce the complexity down to a single, neat expression.

This principle applies universally, so whenever you see a "power of a power," simply multiply the exponents for a quick resolution. It turns a seemingly complex problem into an easy step-by-step journey.

Remember:

• Identify the base and exponents.
• Multiply those exponents together.
• Your final result is a simpler expression, with the base raised to the single, multiplied exponent.

Applying the Product Rule

The product rule simplifies expressions with the same base being multiplied together. It states that when you multiply two expressions with the same base, you add their exponents. This is the next logical step after applying the power rule when required.

For instance, using our previously simplified terms \( t^{12} \) and \( t^{6} \), they share the same base of \( t \). Now, to further simplify, we add the exponents: \( 12 + 6 \) resulting in \( 18 \). Therefore, the expression \( t^{12} \times t^6 \) simplifies nicely to \( t^{18} \).

The beauty of this rule is in its elegance and simplicity:

• Keep the base the same throughout.
• Add together all the exponents attached to that base.

By understanding the product rule, you streamline processes, making the handling of exponential expressions more manageable and less intimidating.

Simplifying Expressions

Simplifying expressions with exponents may initially seem challenging, but by breaking it down using the power and product rules, the task becomes much clearer. Whether dealing with single or multiple terms, your goal is always to reduce the expression to its simplest form.

Let's look back at the original expression \( \left(t^{3}\right)^{4}\left(t^{2}\right)^{3} \). We applied the power rule first to each component, converting this into \( t^{12} \) and \( t^{6} \). This step significantly reduced the complexity of the expression.

Afterward, we employed the product rule, which led to further simplification, resulting finally in \( t^{18} \). This not only makes calculations easier but also helps clarify the problem.

Steps to simplify:

• Use the Power Rule to handle powers raised to powers.
• Use the Product Rule for multiplying like bases.
• Combine these techniques to achieve the simplest expression.

Mastering these rules helps tackle more advanced algebra topics and strengthens problem-solving skills overall.