Simplify (x^(4/7))/(x^(8/7))
The given problem is asking for the simplification of an expression involving two terms with the same base raised to fractional exponents. The expression is a division of one term with the base 'x' raised to the power of 4/7 by another term with the same base 'x' raised to the power of 8/7. To simplify the expression, one needs to use the laws of exponents, specifically the rule that states when dividing powers with the same base, you subtract the exponents. The problem thus requires the application of this rule to obtain the simplest form of the expression.
$\frac{x^{\frac{4}{7}}}{x^{\frac{8}{7}}}$
Answer
Solution:
Step 1:
Apply the rule for negative exponents, which states $a^{-n} = \frac{1}{a^n}$, to rewrite $x^{\frac{4}{7}}$ as a denominator. Thus, we get $\frac{1}{x^{\frac{8}{7}} \cdot x^{-\frac{4}{7}}}$.
Step 2:
Combine the exponents of $x$ by summing them up, as per the exponent addition rule $a^{m} \cdot a^{n} = a^{m+n}$.
Step 2.1:
Using the exponent addition rule, we simplify the expression to $\frac{1}{x^{\frac{8}{7} - \frac{4}{7}}}$.
Step 2.2:
Simplify the exponent by combining the terms over a common denominator, resulting in $\frac{1}{x^{\frac{8 - 4}{7}}}$.
Step 2.3:
Perform the subtraction in the exponent, which gives us $\frac{1}{x^{\frac{4}{7}}}$, and this is our final simplified result.
Knowledge Notes:
Negative Exponent Rule: The negative exponent rule states that for any nonzero number $b$ and any integer $n$, $b^{-n} = \frac{1}{b^n}$. This rule is used to transform expressions with negative exponents into their reciprocal form.
Exponent Addition Rule: When multiplying two expressions with the same base, you can add the exponents. This is formally written as $a^{m} \cdot a^{n} = a^{m+n}$. This rule is essential for simplifying expressions involving powers of the same base.
Combining Fractions: When combining fractions with common denominators, you can add or subtract the numerators while keeping the denominator the same. This principle is used when working with exponents that are expressed as fractions.
Simplification of Expressions: Simplification involves performing arithmetic operations and applying algebraic rules to rewrite an expression in a simpler or more convenient form. In the context of exponents, this often involves adding, subtracting, or otherwise manipulating the exponents to achieve a more simplified expression.
