Simplify (4b)/( fourth root of 7a^3)

Solution:

Step:1

Rationalize the denominator of $\frac{4b}{\sqrt[4]{7a^3}}$ by multiplying by $\frac{\left(\sqrt[4]{7a^3}\right)^3}{\left(\sqrt[4]{7a^3}\right)^3}$.

Step:2

Simplify the expression in the denominator.

Step:2.1

Multiply $\frac{4b}{\sqrt[4]{7a^3}}$ by $\frac{\left(\sqrt[4]{7a^3}\right)^3}{\left(\sqrt[4]{7a^3}\right)^3}$ to get $\frac{4b\left(\sqrt[4]{7a^3}\right)^3}{\left(\sqrt[4]{7a^3}\right)^4}$.

Step:2.2

Express $\sqrt[4]{7a^3}$ as $\left(7a^3\right)^{\frac{1}{4}}$.

Step:2.3

Apply the exponent rule $a^m \cdot a^n = a^{m+n}$.

Step:2.4

Combine exponents to get $\left(\sqrt[4]{7a^3}\right)^4$.

Step:2.5

Recognize that $\left(\sqrt[4]{7a^3}\right)^4$ simplifies to $7a^3$.

Step:3

Simplify the numerator.

Step:3.1

Rewrite $\left(\sqrt[4]{7a^3}\right)^3$ as $\sqrt[4]{\left(7a^3\right)^3}$.

Step:3.2

Raise $7$ to the third power to get $343$.

Step:3.3

Apply the power rule to $a^3$ raised to the third power to get $a^9$.

Step:3.4

Factor $a^8$ from $a^9$ and rewrite it as $\left(a^2\right)^4$.

Step:3.5

Extract $a^2$ from under the fourth root.

Step:4

Reduce the expression by canceling common factors.

Step:4.1

Factor $a^2$ from the numerator.

Step:4.2

Cancel the common $a^2$ factor from the numerator and denominator.

The final simplified expression is $\frac{4b\sqrt[4]{343a}}{7a}$.

Knowledge Notes:

1. Rationalizing the Denominator: This involves removing a radical from the denominator of a fraction by multiplying the numerator and denominator by an appropriate form of 1 that contains the radical.

2. Exponent Rules: These are fundamental in simplifying expressions involving powers. Key rules include:

   • $a^m \cdot a^n = a^{m+n}$
   • $\left(a^m\right)^n = a^{m \cdot n}$
   • $\sqrt[n]{a^m} = a^{\frac{m}{n}}$

3. Simplifying Radicals: When dealing with radicals, we can simplify expressions by extracting factors that are perfect powers of the index of the radical.

4. Factoring: This is the process of breaking down an expression into a product of simpler expressions. In this problem, we factored $a^8$ as $\left(a^2\right)^4$ to simplify the radical.

5. Cancellation: When the same factor appears in both the numerator and the denominator of a fraction, it can be canceled out, simplifying the fraction.

6. Roots and Radicals: The $n$-th root of a number $a$ is written as $\sqrt[n]{a}$ and is equivalent to $a^{\frac{1}{n}}$. When simplifying expressions with roots, we can use this equivalence to rewrite roots as exponents.