Simplify 1/2*(2x^3)^3
The question asks for the simplification of the mathematical expression presented. Specifically, you need to apply the rules of exponents and multiplication to reduce the expression 1/2*(2x^3)^3 to its simplest form.
$\frac{1}{2} \cdot \left(\left(\right. 2 x^{3} \left.\right)\right)^{3}$
Answer
Solution:
Step 1: Apply the exponentiation rule to the expression
Simplify the expression $\frac{1}{2} \cdot (2x^3)^3$ by applying the rule for exponentiation of a product: $(ab)^n = a^n b^n$.
Step 2: Simplify the constants
Identify and simplify the constants in the expression.
Step 2.1: Extract the constant factor from the exponentiated term
Isolate the constant factor $2^3$ from the term $(2x^3)^3$ to get $\frac{1}{2} \cdot 2 \cdot (2^2(x^3)^3)$.
Step 2.2: Eliminate the common factor
Remove the common factor of 2 from the numerator and the constant factor to simplify the expression to $2^2(x^3)^3$.
Step 2.3: Present the simplified constant term
The expression now simplifies to $2^2(x^3)^3$.
Step 3: Simplify the variable expression
Simplify the expression involving the variable $x$.
Step 3.1: Calculate the square of 2
Compute $2^2$ to get 4, resulting in $4(x^3)^3$.
Step 3.2: Apply the power rule to the variable
Apply the power rule to the variable term $(x^3)^3$.
Step 3.2.1: Use the power rule for exponents
Utilize the power rule $(a^m)^n = a^{mn}$ to simplify the variable term to $4x^{3 \cdot 3}$.
Step 3.2.2: Multiply the exponents
Multiply the exponents 3 and 3 to get $4x^9$.
Knowledge Notes:
To solve the given problem, we need to understand several algebraic rules and properties:
Product Rule of Exponents: When two exponential terms with the same base are multiplied, their exponents are added. In this case, we're using the rule in reverse to simplify the expression.
Power Rule of Exponents: When an exponential term is raised to a power, the exponents are multiplied. For example, $(a^m)^n = a^{mn}$.
Simplifying Constants: When dealing with constants, we can perform arithmetic operations like multiplication and division to simplify them.
Canceling Common Factors: When a factor appears in both the numerator and denominator of a fraction, it can be canceled out to simplify the expression.
By applying these rules step by step, we can simplify the given algebraic expression to its simplest form.
