Write with Rational (Fractional) Exponents square root of 5* cube root of 5* fourth root of 5
The given problem is asking for the expression of a product of different roots of the number 5 using rational (fractional) exponents. Specifically, it involves the square root, cube root, and fourth root of the number 5. The task requires the use of exponent rules to combine these roots into a single expression involving powers of 5 with fractional exponents. Each root needs to be translated into its equivalent fractional exponent form before they are multiplied together.
$\sqrt{5} \cdot \sqrt[3]{5} \cdot \sqrt[4]{5}$
Answer
Solution:
Step 1:
Convert $\sqrt{5}$ into exponential form using the rule that $\sqrt[n]{a} = a^{\frac{1}{n}}$. Thus, we get $5^{\frac{1}{2}}$ for the square root of 5.
Step 2:
Similarly, apply the rule to the cube root of 5, which becomes $5^{\frac{1}{3}}$.
Step 3:
Finally, transform the fourth root of 5 into $5^{\frac{1}{4}}$ using the same rule.
Result:
After rewriting all roots in their exponential forms, we have the expression $5^{\frac{1}{2}} \cdot 5^{\frac{1}{3}} \cdot 5^{\frac{1}{4}}$.
Knowledge Notes:
To solve the given problem, we need to understand 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 $\sqrt[n]{a^m}$, where $a$ is the base, $m$ is the exponent, and $n$ is the index of the root.
Exponent Laws: When multiplying expressions with the same base, you add the exponents. For example, $a^x \cdot a^y = a^{x+y}$.
Roots and Radicals: A radical expression is another way to represent a root. The $n$-th root of a number $a$ is written as $\sqrt[n]{a}$ and is equivalent to raising $a$ to the power of $1/n$.
Simplifying Expressions: The process of rewriting expressions in a simpler or more compact form without changing their value.
By applying these concepts, we can rewrite the given expression of multiple roots of the number 5 into an equivalent expression with rational exponents, which can then be further simplified if necessary.
