Write with Rational (Fractional) Exponents (n root of a)^n
The question asks for the expression of a mathematical equation involving roots and exponents using rational (fractional) exponents. Specifically, it refers to converting the n-th root of a variable 'a' raised to the n-th power into an equivalent expression using exponents expressed as fractions rather than using radical symbols. It involves applying rules of exponents and understanding the relationship between roots and exponents to rewrite the expression in a simpler or alternative form.
$\left(\left(\right. \sqrt[n]{a} \left.\right)\right)^{n}$
Answer
Solution:
Step 1:
Convert the n-th root of a to an exponent by using the identity $\sqrt[n]{a} = a^{\frac{1}{n}}$. Thus, we have $\left(\sqrt[n]{a}\right)^{n} = \left(a^{\frac{1}{n}}\right)^{n}$.
Step 2:
Proceed to multiply the exponents in the expression $\left(a^{\frac{1}{n}}\right)^{n}$.
Step 2.1:
Utilize the rule of exponents that states $\left(a^{m}\right)^{n} = a^{mn}$. This gives us $a^{\frac{1}{n} \cdot n}$.
Step 2.2:
Eliminate the common n in the numerator and the denominator.
Step 2.2.1:
Simplify by removing the common factor: $a^{\frac{1}{\cancel{n}} \cdot \cancel{n}}$.
Step 2.2.2:
Express the simplified form: $a^{1}$, which is simply $a$.
Step 3:
The final simplification yields $a$.
Knowledge Notes:
To understand the solution, it's essential to be familiar with the following concepts:
Rational (Fractional) Exponents: A rational exponent represents both an exponent and a root. The expression $a^{\frac{m}{n}}$ is equivalent to the n-th root of $a$ raised to the m-th power, or $\sqrt[n]{a^{m}}$.
n-th Root: The n-th root of a number a, written as $\sqrt[n]{a}$, is a number that, when raised to the power of n, equals a.
Exponent Rules: There are several rules for manipulating exponents, including:
Power Rule: $\left(a^{m}\right)^{n} = a^{mn}$, which states that when raising a power to another power, you multiply the exponents.
Product Rule: $a^{m} \cdot a^{n} = a^{m+n}$, which states that when multiplying like bases, you add the exponents.
Quotient Rule: $\frac{a^{m}}{a^{n}} = a^{m-n}$, which states that when dividing like bases, you subtract the exponents.
Simplification: The process of reducing an expression to its simplest form. In the context of exponents, this often involves applying the exponent rules to combine or reduce terms.
By applying these concepts, we can rewrite expressions with rational exponents in a simpler form or with integer exponents, which can be easier to work with in further calculations.
