Question

By giving specific values for \(a, b,\) and \(c,\) explain how the exponent rule $$ \left(a^{b}\right)^{c}=a^{b c} $$ is used to rewrite the expressions in Problems \(39-40 .\) $$ \left(2 m^{2} n^{4}\right)^{3 r+3}=\left(8 m^{6} n^{12}\right)^{r+1} $$

Short answer

Answer: Yes, the expressions $$\left(2 m^{2} n^{4}\right)^{3 r+3}$$ and $$\left(8 m^{6} n^{12}\right)^{r+1}$$ are equivalent using the exponent rule $$\left(a^{b}\right)^{c}=a^{b c}$$.

Step-by-step solution

Understand the exponent rule with specific values

Let's understand the given exponent rule, $$\left(a^{b}\right)^{c}=a^{b c}$$, with specific values of \(a\), \(b\), and \(c\). For example, let a=2, b=3, and c=4, then: $$ \left(2^{3}\right)^{4}=\left(2^3\right)^4=(8)^4=8^{3\cdot4}=2^{12} $$

Apply the exponent rule to the expressions

Now, we will apply the exponent rule to rewrite the given expressions: $$\left(2 m^{2} n^{4}\right)^{3 r+3}=\left(8 m^{6} n^{12}\right)^{r+1}$$ As we can see, the given expression involves variables \(m\) and \(n\). We can apply the exponent rule to each of the variables separately. Let's start with the left-hand side (LHS): $$ \left(2 m^{2} n^{4}\right)^{3 r+3}= 2^{3 r+3} \cdot \left(m^{2}\right)^{3 r+3} \cdot \left(n^{4}\right)^{3 r+3} $$ Now, apply the exponent rule to each part: $$ 2^{3 r+3} \cdot m^{2(3 r+3)} \cdot n^{4(3 r+3)} = 2^{3 r+3} \cdot m^{6 r+6} \cdot n^{12 r+12} $$ Next, move to the right-hand side (RHS): $$ \left(8 m^{6} n^{12}\right)^{r+1} = 8^{r+1} \cdot \left(m^{6}\right)^{r+1} \cdot \left(n^{12}\right)^{r+1} $$ Now, apply the exponent rule to each part: $$ 8^{r+1} \cdot m^{6(r+1)} \cdot n^{12(r+1)} = 2^{3(r+1)} \cdot m^{6 r+6} \cdot n^{12 r+12} $$

Compare the rewritten expressions

Now, we can compare the rewritten expressions of the LHS and RHS to see if they are equal: LHS: $$2^{3 r+3} \cdot m^{6 r+6} \cdot n^{12 r+12}$$ RHS: $$2^{3(r+1)} \cdot m^{6 r+6} \cdot n^{12 r+12}$$ As we can see, the rewritten expressions for both the LHS and RHS are equal. Hence, the given expressions are indeed equivalent by using the exponent rule $$\left(a^{b}\right)^{c}=a^{b c}$$.

Key concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is foundational for more complex mathematics and deals primarily with expressions, equations, and functions.
In this exercise, algebra plays a crucial role in using the exponent rules to simplify expressions. We start by breaking down expressions into parts that are easier to manage or compare. This involves recognizing common bases or patterns in terms and applying algebraic principles to find equalities or reduce complexity.

• Understanding the basic laws of algebra is key to manipulating expressions.
• Algebraic expressions can contain numbers, variables, and operations that together represent a particular value or relationship.
• By systematically applying algebraic techniques, such as distributive, commutative, and associative properties, one can simplify expressions and solve equations more efficiently.

In this problem, algebraic techniques help us translate a complex exponent expression into an equivalent, simplified form by leveraging multiplication within powers.

Expression Simplification

Expression simplification in algebra involves reducing the complexity of expressions while preserving their value. The main goal is to rewrite expressions in a simpler or more easily understandable form.
In this exercise, simplifying an expression involves using the exponent rule \(\left(a^{b}\right)^{c} = a^{bc}\). By applying this rule, you can become more efficient at controlling how terms interact with each other.

• The power of a product rule allows you to apply an exponent to each term within a grouping separately. This is key when dealing with expressions like \(\left(2m^{2}n^{4}\right)^{3r+3}\).
• Separate expressions into base and exponent, apply the rule to change the expression from nested exponents to a single exponent term.
• This reduces the expression to terms where each variable or number has a single exponent, making it easier to compare or combine with other expressions.

Simplifying expressions this way not only makes them more elegant but also simplifies further operations, such as addition, subtraction, or substitution.

Equality of Expressions

Equality of expressions refers to two algebraic expressions representing the same value, even if they appear different at first glance. Achieving this often involves simplifying or rearranging one or both expressions.
In this exercise, we used algebraic manipulation to show that two different-looking expressions are indeed equal by rewriting them using the exponent rule \(\left(a^{b}\right)^{c} = a^{bc}\). This rule allows us to compress and expand expressions in a controlled manner.

• We verify equality by ensuring every term on one side matches with terms on the other side after simplification.
• Both the left-hand side and the right-hand side share a common structure after applying the rule.
• The bases and their corresponding exponents must be equivalent for the entire expressions to be considered identical.

By demonstrating their equality, we ensure that we maintain balance and correctness in equations, which is fundamental to solving algebraic problems. The consideration of equal expressions also illustrates how seemingly different algebraic forms can represent the same mathematical truths.