Write with Rational (Fractional) Exponents cube root of 5^5* square root of 5
In the given problem, you are asked to express a mathematical expression involving cube root and square root operations on the number 5 raised to a specific power, using rational (or fractional) exponents instead of radical symbols. Rational exponents denote roots and powers by using fractions, where the numerator indicates the power and the denominator indicates the root being taken. The problem seeks a rewritten form of the expression that aligns with the conventions of using exponents rather than radical notation.
$\sqrt[3]{5^{5}} \cdot \sqrt{5}$
Answer
Solution:
Step 1:
Convert the cube root of $5^5$ into an expression with a rational exponent using the formula $a^{\frac{x}{n}}$ for $\sqrt[n]{a^x}$. This gives us $5^{\frac{5}{3}}$.
Combine this with the square root of 5 to get $5^{\frac{5}{3}} \cdot \sqrt{5}$.
Step 2:
Similarly, express the square root of 5 as a power with a fractional exponent, which is $5^{\frac{1}{2}}$. Now we have $5^{\frac{5}{3}} \cdot 5^{\frac{1}{2}}$.
Knowledge Notes:
To solve problems involving roots and exponents, it's often helpful to use the property of exponents that allows us to express roots as rational (fractional) exponents. The general formula is:
$$\sqrt[n]{a^x} = a^{\frac{x}{n}}$$
where $a$ is the base, $x$ is the exponent, and $n$ is the degree of the root. This formula is derived from the definition of rational exponents and the properties of exponents. It's particularly useful because it allows us to work with roots in the same way we work with exponents, making it easier to multiply and divide expressions with different roots.
Another important property of exponents used in this problem is that when you multiply expressions with the same base, you can add the exponents:
$$a^m \cdot a^n = a^{m+n}$$
This property is essential when combining terms with the same base but different exponents, as seen in the problem where we combine $5^{\frac{5}{3}}$ and $5^{\frac{1}{2}}$.
