Simplify (x^(1/3))/(x^(1/12))
The given question is asking to simplify an expression involving powers of the variable x. Specifically, the expression provided is in the form of a fraction where the numerator is x raised to the power of one-third, and the denominator is x raised to the power of one-twelfth. The objective is to apply the rules of exponents to this expression to simplify it to its most reduced form, by combining the exponents in an appropriate manner, given that they have the same base 'x'.
$\frac{x^{\frac{1}{3}}}{x^{\frac{1}{12}}}$
Answer
Solution:
Step 1:
Apply the negative exponent rule to bring $x^{\frac{1}{12}}$ to the numerator, which states $\frac{1}{b^n} = b^{-n}$. Thus, we get $x^{\frac{1}{3}} \cdot x^{-\frac{1}{12}}$.
Step 2:
Combine the exponents of $x$ by adding them together.
Step 2.1:
Utilize the exponent addition rule $a^m \cdot a^n = a^{m+n}$ to simplify the expression to $x^{\frac{1}{3} - \frac{1}{12}}$.
Step 2.2:
Convert $\frac{1}{3}$ to an equivalent fraction with a denominator of $12$ by multiplying by $\frac{4}{4}$, resulting in $x^{\frac{4}{12} - \frac{1}{12}}$.
Step 2.3:
Ensure both fractions have a common denominator of $12$.
Step 2.3.1:
Multiply $\frac{1}{3}$ by $\frac{4}{4}$ to get $\frac{4}{12}$.
Step 2.3.2:
Now we have $x^{\frac{4}{12} - \frac{1}{12}}$.
Step 2.4:
Add the numerators over the common denominator to get $x^{\frac{4 - 1}{12}}$.
Step 2.5:
Subtract the numerators to simplify to $x^{\frac{3}{12}}$.
Step 2.6:
Reduce the fraction by eliminating common factors between the numerator and the denominator.
Step 2.6.1:
Factor out a $3$ from the numerator to get $x^{\frac{3 \cdot 1}{12}}$.
Step 2.6.2:
Eliminate the common factors between the numerator and the denominator.
Step 2.6.2.1:
Factor out a $3$ from the denominator to get $x^{\frac{3 \cdot 1}{3 \cdot 4}}$.
Step 2.6.2.2:
Cancel out the common factor of $3$ to simplify to $x^{\frac{\cancel{3} \cdot 1}{\cancel{3} \cdot 4}}$.
Step 2.6.2.3:
Finally, rewrite the expression as $x^{\frac{1}{4}}$.
Knowledge Notes:
The problem-solving process involves simplifying an expression with exponents by using exponent rules. The relevant knowledge points include:
Negative exponent rule: $\frac{1}{b^n} = b^{-n}$. This rule allows us to move a factor from the denominator to the numerator by changing the sign of its exponent.
Power rule for exponents: $a^m \cdot a^n = a^{m+n}$. When multiplying expressions with the same base, we add their exponents.
Common denominator: When combining fractions, we need to ensure they have a common denominator before we can add or subtract the numerators.
Reducing fractions: If the numerator and denominator of a fraction have a common factor, we can cancel it out to simplify the fraction.
Simplification: The goal is to express the result in the simplest form, which often means reducing fractions to their lowest terms.
