Problem 132 - College Algebra

Question

In Exercises 132–135, determine whether each statement makes sense or does not make sense, and explain your reasoning. There are many exponential expressions that are equal to \(36 x^{12},\) such as \(\left(6 x^{6}\right)^{2},\left(6 x^{3}\right)\left(6 x^{9}\right), 36\left(x^{3}\right)^{9},\) and \(6^{2}\left(x^{2}\right)^{6}\)

Step-by-Step Solution

Answer

The expressions \((6x^{6})^{2}\), \(6x^{3}*(6x^{9})\) and \(6^2 * (x^{2})^{6}\) are equals to \(36x^{12}\), while \(36*(x^{3})^{9}\) is not.

Step 1: Evaluate (6x^6)^2

In this exponential expression, exponentiation of a product rule applies, which states that, for any numbers a and b, and integer n: \((ab)^n = a^n * b^n\). Therefore, \((6 x^{6})^{2} = 6^2 * (x^{6})^{2} = 36x^{12}\) which is equal to the given expression \(36x^{12}\).

Step 2: Evaluate \(6x^3(6x^9)\)

In multiplication, exponents can be added together when the base is the same. So by that rule \(6x^3*6x^9 = 36x^{3+9}\ = 36x^{12}\), which is also equal to the given expression.

Step 3: Evaluate \(36(x^3)^9\)

Remember how to exponentiate a power, which states that, for any number a, and integers m and n, \((a^m)^n = a^{m*n}\), this gives \((36*(x^3)^9 = 36*x^{27}\), this is not equal to the given expression.

Step 4: Evaluate \(6^2(x^2)^6\)

Applying the rule of exponentiating a power again, \(6^2*(x^2)^6 = 36*x^{12}\), this matches the given expression.

Key Concepts

Exponentiation of a Product Rule

Understanding the exponentiation of a product rule is pivotal when dealing with exponential expressions. This rule simply states that when you have an expression in the form of \( (ab)^n \), you can apply the exponent to both factors separately: \( (ab)^n = a^n * b^n \).

In the context of the given exercise, when you have \( (6x^6)^2 \), it's the same as saying \( 6^2 \times (x^6)^2 \). This simplifies to \( 36x^{12} \) because each term in the product is raised to the power of 2. This rule helps streamline simplifying expressions, turning what might seem complex into an easily solvable problem.

The key here is recognizing that the exponent outside the parentheses applies to every element within the parentheses, and this method can be applied each time you encounter a product raised to a power.

Addition of Exponents Rule

When multiplying exponents with the same base, the 'addition of exponents' rule comes into play. For expressions like \( a^m * a^n \), the rule tells us to simply add the exponents: \( a^m * a^n = a^{m+n} \).

In our exercise, we looked at the expression \( 6x^3(6x^9) \). Since we have the same base \( x \), we can add the exponents directly: \( 6x^3 * 6x^9 = 36x^{3+9} \), which simplifies to \( 36x^{12} \) – matching the original expression perfectly. It's essential to note that this rule is only applicable when the bases are the same. If the bases were different, addition would not be the same.

Exponentiating a Power

For expressions where a power is raised to another power, like \( (a^m)^n \), the rule is to multiply the exponents: \( (a^m)^n = a^{m*n} \). This is known as 'exponentiating a power'.

Consider the expression \( 36(x^3)^9 \) from the exercise. Applying our rule, we would multiply the exponents, giving us \( 36x^{3*9} \) which equates to \( 36x^{27} \). It's different from the target expression \( 36x^{12} \), showing a clear discrepancy. This example serves as a reminder to apply this rule carefully and to always check the resulting expression against the original to ensure accuracy.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They are fundamental in representing relationships between quantities and solving a wide array of problems.

The beauty of algebraic expressions lies in their ability to be manipulated through simplification, expansion, and application of various algebraic rules. Each step in our exercise has shown how, by understanding these rules, complex expressions can be broken down and transformed into simpler, equivalent forms.

For instance, each rule we've applied in previous sections, whether it was the 'exponentiation of a product rule', the 'addition of exponents rule', or the 'exponentiating a power', turned each complicated expression into the simple form \( 36x^{12} \) - with one exception that served as a cautionary tale. This exemplifies the power and utility of algebraic expressions when embarking on the journey of solving mathematical problems.