Problem 21 - Algebra Form and Function

Question

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \left(3 x^{2}\right)^{3} $$

Step-by-Step Solution

Answer

If yes, identify the values of \(k\) and \(p\). Answer: Yes, the expression can be written in the form \(kx^p\). The values are \(k = 27\) and \(p = 6\).

Step 1: Simplify the expression by using exponent rules

We can use the power of a power rule \((a^m)^n = a^{mn}\). Applying this rule, we get: $$ \left(3 x^{2}\right)^{3} = 3^{3}(x^2)^3 $$

Step 2: Simplify the expression further

We can now calculate \(3^3\) and apply the power of a power rule to the \(x\) term as well: $$ 3^{3}(x^2)^3 = 27x^{2\cdot 3} $$

Step 3: Write the simplified expression in the form \(kx^p\)

We now write the expression in the form \(kx^p\) by calculating the exponent for \(x\): $$ 27x^{2\cdot 3} = 27x^6 $$ This expression is now in the form \(kx^p\). From this expression, we can see that:

Step 4: Identifying the values of \(k\) and \(p\)

Here are the values of \(k\) and \(p\): $$ k = 27 \\ p = 6 $$ So, yes, the expression can be written in the form \(kx^p\), and the values are \(k = 27\) and \(p = 6\).

Key Concepts

Exponent Rules

Exponent rules are essential guidelines in algebra for performing operations involving powers of numbers or variables. They allow us to simplify expressions and solve equations efficiently. One of the fundamental exponent rules is the product of powers rule, which states that when multiplying like bases, we add the exponents: \[a^m \cdot a^n = a^{m+n}\] This rule helps condense expressions like \(x^3 \cdot x^4\) into a simpler form: \(x^{3+4} = x^7\).
The quotient of powers rule is another handy rule. It indicates we subtract the exponents when dividing like bases: \[\frac{a^m}{a^n} = a^{m-n}\] For example, \(\frac{x^5}{x^2} = x^{5-2} = x^3\). These rules streamline our calculations and should be ingrained into any student's toolkit. If handled well, exponent rules can make complex expressions much more straightforward.

Power of a Power Rule

The power of a power rule is a specific exponent rule particularly useful when dealing with expressions in which a power is raised to another power. The formula for this rule is: \[(a^m)^n = a^{mn}\] This means we multiply the exponents together to simplify the expression. For instance, with \((x^2)^3\), we apply the rule directly to get \(x^{2 \times 3} = x^6\).
In our original problem, \((3x^2)^3\), the rule is applied by treating each component separately:

• For the coefficient \(3\), you calculate \(3^3\) to get \(27\).
• For the variable part \((x^2)^3\), use the power of a power rule to simplify to \(x^6\).

This results in the expression being \(27x^6\). Understanding this rule can greatly simplify the handling of multiple layers of exponents in a given expression.

Simplifying Expressions

Simplifying expressions is a critical skill in algebra where we reduce expressions to their simplest form without changing their value. It involves applying mathematical rules, like exponent rules, to make expressions easier to read and work with.
Take our original example: starting with \((3x^2)^3\). Employ the power of a power rule to individual parts, calculating \(3^3\) to get \(27\) and \((x^2)^3\) giving \(x^6\). After applying these transformations, the expression becomes \(27x^6\).

• Simplification makes the expression easier to understand and evaluate.
• It often involves combining like terms, reducing fractions, and eliminating unnecessary parentheses.

Simplifying complex expressions like in this exercise is not only satisfying but also plays a crucial role in advancing mathematical problem-solving skills. Mastery in simplification leads to more efficiency when working through equations in algebra and helps build a deeper understanding of mathematical structures.