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Magazine Chapter 0: Problem 24
Simplify each expression. $$ \frac{1}{6} x\left(3 x^{2}\right)^{3} $$
Short Answer
Expert verified
\( \frac{9}{2}x^7 \)
Step by step solution
01
Expand the Exponent
First, recognize that the expression inside the parentheses, \((3x^2)^3\), has an exponent, so expand it. To do this, apply the power of a power rule, \((a^m)^n = a^{m imes n}\). Here, this rule applies in such a way that the base \(3x^2\) is raised to the 3rd power.
02
Apply the Power of a Power Rule
Using the power of a power rule, calculate \((3x^2)^3\). This results in \(3^3 (x^2)^3\) or \(27x^6\). Thus, the expression becomes: \[ \frac{1}{6} x(27x^6) \]
03
Distribute the Multiplication
Now distribute \(x\) to the expanded expression \(27x^6\). This is done by multiplying the coefficients and applying the product of powers rule \(x^m \cdot x^n = x^{m+n}\). So, we have: \[ \frac{1}{6} imes 27 imes x^{1+6} \] This simplifies the expression to: \[ \frac{27}{6} x^7 \]
04
Simplify the Coefficient
The next step is to simplify the fraction \(\frac{27}{6}\). To simplify, find the greatest common divisor of 27 and 6, which is 3. Divide both the numerator and the denominator by 3: \[ \frac{27}{6} = \frac{9}{2} \]
05
Combine Terms for Final Expression
Finally, combine the simplified fraction with the \(x\) terms. The complete simplified expression becomes: \[ \frac{9}{2}x^7 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Simplification
Simplification is the process of reducing an expression to its most concise form. This involves combining like terms, reducing fractions, and applying rules of arithmetic to make the expression easier to understand or solve.
Ultimately, simplification not only makes expressions easier to read but also often further facilitates problem-solving or equation-solving processes.
Consider the expression: \( \frac{1}{6} x (3x^2)^3 \). Our task is to simplify this into a more straightforward expression.
- The first step is to deal with the exponentiation within the parentheses, which means we will use specific exponent rules.
- The product, \( (3x^2)^3 \), includes both a coefficient and a variable raised to a power.
Ultimately, simplification not only makes expressions easier to read but also often further facilitates problem-solving or equation-solving processes.
Power of a Power Rule
The power of a power rule is a central rule in exponentiation. It states that when an expression involving an exponent itself is raised to another power, you multiply the exponents.
For example, consider \( (a^m)^n \); according to the rule, this will be equal to \( a^{m \times n} \).
This rule is crucial whenever dealing with nested powers, often appearing in algebraic manipulations.
For example, consider \( (a^m)^n \); according to the rule, this will be equal to \( a^{m \times n} \).
In our example, \( (3x^2)^3 \) involves applying this rule:
- Calculate: \( (x^2)^3 = x^{2 \times 3} = x^6 \)
- For the whole expression, \( 3^3 = 27 \), then combine the results to get \( 27x^6 \).
This rule is crucial whenever dealing with nested powers, often appearing in algebraic manipulations.
Product of Powers Rule
The product of powers rule is another key principle of exponentiation. It provides a method to combine like bases with exponents. This rule declares that when multiplying two expressions with the same base, their exponents should be added.
The rule can be presented as: \( a^m \times a^n = a^{m+n} \).
Utilizing the product of powers rule is essential in algebra to simplify expressions, making them easier to work with and solve.
The rule can be presented as: \( a^m \times a^n = a^{m+n} \).
In the expression we are simplifying, once we have \( 27x^6 \) from the previous step, we need to multiply it with \( x \):
- Identify the same base (which is \( x \)), then apply the rule: \( x^1 \times x^6 = x^{1+6} = x^7 \).
Utilizing the product of powers rule is essential in algebra to simplify expressions, making them easier to work with and solve.
