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Chapter 0: Problem 24

simplify the expression. $$ \left(4 x^{3}\right)^{2} $$

Short Answer

Expert verified
The simplified expression is \( 16x^6 \).

Step by step solution

01

Apply the power of a power rule

Applying the power of a power rule which states that \( (a^m)^n = a^{m*n} \), we get \( (4x^3)^2 = 4^2 * (x^3)^2 \).
02

Simplify the expression

Evaluating the expression from the Step 1, we get \( 4^2 * x^{3*2} = 16 * x^6 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power of a Power Rule
The Power of a Power Rule is a fundamental principle when working with exponents. This rule tells us that when we raise a power to another power, we multiply the exponents. It can be illustrated as follows:
  • If you have an expression \( (a^m)^n \), it simplifies to \( a^{m \times n} \).
  • This rule significantly simplifies working with expressions that have repeated exponents.
  • It helps to transform seemingly complex expressions into simpler forms.
For instance, in the expression \((4x^3)^2\), applying the Power of a Power Rule allows us to rewrite it as \(4^2\) multiplied by \((x^3)^2\). Essentially, we distribute the outside exponent to both the base and the inner exponent, making the computation more straightforward.
Simplification
Simplification is the process of reducing an expression to its simplest form. This makes it easier to understand or work with, especially in complicated algebraic problems.
  • A simpler form is often one with fewer terms or a single term with lower or equal exponents.
  • We achieve this by applying rules of operations and properties of exponents effectively.
In our example, after applying the Power of a Power Rule, we further simplify \(4^2\) to 16, and \((x^3)^2\) to \(x^6\). Thus, the expression \((4x^3)^2\) simplifies to \(16 \cdot x^6\). Remember, simplification often involves combining like terms and calculating any constants as much as possible.
Exponents
Exponents express repeated multiplication of a number by itself. They form a compact form of writing numbers that have large multiplicative factors.
  • The number itself is the "base" and the small raised number is the "exponent".
  • An exponent of 2, for example, means the base is multiplied by itself once (\(x^2 = x \, \cdot \, x\)).
  • Working with exponents involves applying various rules, such as the addition, subtraction, and multiplication of exponents.
These rules allow us to simplify expressions efficiently. For example, in \((x^3)^2\), you multiply the exponents (3 and 2), resulting in \(x^6\). Efficient knowledge of exponent rules is key to simplifying algebraic expressions properly.
Polynomial Functions
Polynomial functions are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
  • A simple example would be \(3x^2 + 4x + 5\).
  • They are important in algebra as they appear in equations and allow us to model a wide variety of real-world phenomena.
  • When we simplify expressions like \((4x^3)^2\), we're working towards identifying or manipulating polynomial expressions.
In the simplified expression \(16 \cdot x^6\), we have a single term polynomial. Understanding polynomial functions is crucial as it aligns with various algebraic tasks, from simple equations to complex financial and scientific modelling.

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Most popular questions from this chapter

a certificate of deposit has a principal of P and an annual percentage rate of r (expressed as a decimal) compounded n times per year. Enter the compound interest formula $$ A=P\left(1+\frac{r}{n}\right)^{N} $$ into a graphing utility and use it to find the balance after \(N\) compoundings. $$ P=\$ 7000, \quad r=5 \%, \quad n=365, \quad N=1000 $$

simplify each expression by factoring. $$ 2 x(x-1)^{5 / 2}-4(x-1)^{3 / 2} $$

evaluate the expression for the given value of x. $$ \frac{1+x^{-1}}{x^{-1}} \quad x=3 $$

evaluate the expression for the given value of x. $$ 500 x^{60} \quad x=1.01 $$

Rationalize the numerator or denominator and simplify. $$ \frac{10(x+2)}{\sqrt{x^{2}-x-6}} $$
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