Problem 62 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( r^{14} \).

Step 1: Apply the Product Rule for Exponents Inside the Parenthesis

The product rule for exponents states that when multiplying two powers with the same base, you can add the exponents: \( a^m \cdot a^n = a^{m+n} \). In this expression, apply it inside the parentheses: \( r^3 \cdot r^4 = r^{3+4} = r^7 \). So, the expression becomes \( (r^7)^2 \).

Step 2: Apply the Power Rule for Exponents

The power rule for exponents states that when raising a power to another power, you multiply the exponents: \( (a^m)^n = a^{m\cdot n} \). Apply this to \( (r^7)^2 \): \( r^{7\cdot 2} = r^{14} \).

Key Concepts

Product Rule for Exponents

The product rule for exponents is incredibly useful when dealing with expressions involving the multiplication of variables raised to powers. This rule tells us what to do when we have two terms with the same base being multiplied together. Here's a simple way to remember the rule:

• If you multiply terms with the same base, simply add their exponents.

In mathematical terms, if you have an expression such as \( a^m \times a^n \), you can simplify it to \( a^{m+n} \).

Consider the example \( r^3 \times r^4 \). Both terms have the base \( r \). So, the exponents \( 3 \) and \( 4 \) are added together, resulting in \( r^{3+4} = r^7 \). This step simplifies our work by condensing the expression within the parentheses.

Power Rule for Exponents

The power rule for exponents is another handy shortcut for managing expressions involving powers raised to other powers. This rule helps us handle expressions where there is one power within another. It's like a way to simplify nested exponents.

Here's the rule:

• When you have a power raised to a power, multiply the exponents.

If you see \( (a^m)^n \), you can simplify it into \( a^{m \cdot n} \).

In our example, after using the product rule, we ended up with \( (r^7)^2 \). To simplify it further using the power rule, multiply the exponents: \( 7 \times 2 \), giving us \( r^{14} \). Each step using these rules brings us closer to a more manageable expression.

Simplifying Expressions

Simplifying expressions involves combining like terms and applying mathematical rules to make expressions shorter and easier to understand or work with. This is an essential skill in algebra, as it reduces complexity and allows you to solve problems more efficiently.

When simplifying expressions using exponent rules, your goal is to apply the appropriate rules systematically. In our case, we start with \( (r^3 \times r^4)^2 \).

You might break it down as follows:

• First, observe the bases of the terms you need to simplify.
• Apply the product rule to combine exponents within the parentheses.
• Then apply the power rule to simplify the result.
• This results in the final simplified expression: \( r^{14} \).

Each of these steps uses the rules of exponents to transform the original, more complicated expression into a simple one, making algebraic operations smoother and more intuitive.