Simplify (4x^2*(2x^2))/(12x*(3x))
The problem is asking for a simplification of a given algebraic expression which involves multiplication and division of polynomials.
The expression contains variables (x) raised to certain exponents, and numerical coefficients. The task involves applying the rules of exponents, simplifying the numeric coefficients through division, and cancelling out any common factors within the numerator and the denominator.
The challenge is to reduce the expression to its simplest form by performing the operations indicated and following the proper order of operations for algebraic expressions.
$\frac{4 x^{2} \cdot \left(\right. 2 x^{2} \left.\right)}{12 x \cdot \left(\right. 3 x \left.\right)}$
Answer
Solution:
Step 1
Combine the powers of $x$ in the numerator using exponent addition.
Step 1.1
Consider the expression $\frac{4(x^2 \cdot x^2) \cdot 2}{12x \cdot (3x)}$.
Step 1.2
Apply the exponent rule $a^m \cdot a^n = a^{m+n}$ to merge the exponents: $\frac{4x^{2+2} \cdot 2}{12x \cdot (3x)}$.
Step 1.3
Perform the addition within the exponent: $\frac{4x^4 \cdot 2}{12x \cdot (3x)}$.
Step 2
Combine the powers of $x$ in the denominator using exponent addition.
Step 2.1
Rewrite the denominator as $12(x \cdot x) \cdot 3$: $\frac{4x^4 \cdot 2}{12(x \cdot x) \cdot 3}$.
Step 2.2
Apply the exponent rule to the denominator: $\frac{4x^4 \cdot 2}{12x^2 \cdot 3}$.
Step 3
Reduce the fraction by eliminating common factors.
Step 3.1
Extract the common factor of 4 from the numerator: $\frac{4(x^4 \cdot 2)}{12x^2 \cdot 3}$.
Step 3.2
Eliminate common factors between the numerator and denominator.
Step 3.2.1
Extract the common factor of 4 from the denominator: $\frac{4(x^4 \cdot 2)}{4(3x^2 \cdot 3)}$.
Step 3.2.2
Cancel out the common factor of 4: $\frac{\cancel{4}(x^4 \cdot 2)}{\cancel{4}(3x^2 \cdot 3)}$.
Step 3.2.3
Simplify the expression: $\frac{x^4 \cdot 2}{3x^2 \cdot 3}$.
Step 4
Reduce the powers of $x$ by canceling out common factors.
Step 4.1
Factor out $x^2$ from $x^4 \cdot 2$: $\frac{x^2(x^2 \cdot 2)}{3x^2 \cdot 3}$.
Step 4.2
Eliminate common $x^2$ factors.
Step 4.2.1
Factor out $x^2$ from $3x^2 \cdot 3$: $\frac{x^2(x^2 \cdot 2)}{x^2(3 \cdot 3)}$.
Step 4.2.2
Cancel out the common $x^2$ factor: $\frac{\cancel{x^2}(x^2 \cdot 2)}{\cancel{x^2}(3 \cdot 3)}$.
Step 4.2.3
Simplify the expression: $\frac{x^2 \cdot 2}{3 \cdot 3}$.
Step 5
Final simplification of the expression.
Step 5.1
Multiply the constants in the denominator: $\frac{x^2 \cdot 2}{9}$.
Step 5.2
Reposition the constant in the numerator: $\frac{2x^2}{9}$.
Knowledge Notes:
This problem involves simplifying a rational expression with polynomial terms. The key knowledge points include:
Exponent Rules: When multiplying like bases, the exponents are added together. The power rule states that $a^m \cdot a^n = a^{m+n}$.
Reducing Fractions: To simplify a fraction, common factors in the numerator and denominator can be canceled out. This includes both numerical coefficients and variables with exponents.
Factoring: Extracting common factors from terms simplifies the expression and makes it easier to identify and cancel out common terms.
Simplifying Expressions: The final step in simplifying a rational expression is to ensure that all like terms have been combined and the expression is in its simplest form.
By applying these principles, the given expression is simplified step by step, resulting in a more concise and manageable form.
