Simplify cube root of 3/(5a^4)

The question is asking to take the expression that involves a cube root and simplify it. Specifically, it involves the cube root of a fraction where the numerator is the value 3, and the denominator is the product of the number 5 and the variable 'a' raised to the fourth power, denoted as 5a^4. Simplifying this expression would usually involve rewriting it in a form that is potentially easier to work with or understand, often by reducing any exponents and removing the cube root if possible.

$\sqrt[3]{\frac{3}{5 a^{4}}}$

Answer

Expert–verified

Solution:

Step 1

Express $\frac{3}{5 a^{4}}$ as ${(\frac{1}{a})}^{3} \cdot \frac{3}{5 a}$ .

Step 1.1

Extract the cube of $1$ from the numerator: $\sqrt[3]{\frac{1^{3} \cdot 3}{5 a^{4}}}$ .

Step 1.2

Extract the cube of $a$ from the denominator: $\sqrt[3]{\frac{1^{3} \cdot 3}{a^{3} \cdot (5 a)}}$ .

Step 1.3

Reorganize the fraction as $\sqrt[3]{{(\frac{1}{a})}^{3} \cdot \frac{3}{5 a}}$ .

Step 2

Remove terms from under the cube root: $\frac{1}{a} \cdot \sqrt[3]{\frac{3}{5 a}}$ .

Step 3

Decompose the cube root: $\frac{1}{a} \cdot \frac{\sqrt[3]{3}}{\sqrt[3]{5 a}}$ .

Step 4

Rationalize the denominator by multiplying by $\frac{(\sqrt[3]{5 a})^{2}}{(\sqrt[3]{5 a})^{2}}$ .

Step 5

Simplify the denominator.

Step 5.1

Multiply the terms: $\frac{1}{a} \cdot \frac{\sqrt[3]{3} \cdot (\sqrt[3]{5 a})^{2}}{\sqrt[3]{5 a} \cdot (\sqrt[3]{5 a})^{2}}$ .

Step 5.2

Raise $\sqrt[3]{5 a}$ to the power of $1$ : $\frac{1}{a} \cdot \frac{\sqrt[3]{3} \cdot (\sqrt[3]{5 a})^{2}}{(\sqrt[3]{5 a})^{1 + 2}}$ .

Step 5.3

Combine exponents using the power rule $a^{m} \cdot a^{n} = a^{m + n}$ .

Step 5.4

Sum the exponents: $\frac{1}{a} \cdot \frac{\sqrt[3]{3} \cdot (\sqrt[3]{5 a})^{2}}{(\sqrt[3]{5 a})^{3}}$ .

Step 5.5

Convert $(\sqrt[3]{5 a})^{3}$ back to $5 a$ .

Step 5.5.1

Rewrite $\sqrt[3]{5 a}$ as $(5 a)^{\frac{1}{3}}$ .

Step 5.5.2

Apply the power rule $(a^{m})^{n} = a^{m n}$ .

Step 5.5.3

Multiply $\frac{1}{3}$ by $3$ .

Step 5.5.4

Cancel the common factor of $3$ .

Step 5.5.5

Simplify the expression to $\frac{1}{a} \cdot \frac{\sqrt[3]{3} \cdot (\sqrt[3]{5 a})^{2}}{5 a}$ .

Step 6

Combine the terms: $\frac{\sqrt[3]{3} \cdot (\sqrt[3]{5 a})^{2}}{a \cdot 5 a}$ .

Step 7

Add the exponents of $a$ .

Step 7.1

Rearrange $a$ : $\frac{\sqrt[3]{3} \cdot (\sqrt[3]{5 a})^{2}}{a^{2} \cdot 5}$ .

Step 7.2

Multiply $a$ by $a$ .

Step 8

Multiply $\sqrt[3]{3} \cdot (\sqrt[3]{5 a})^{2}$ by $1$ .

Step 9

Simplify the numerator.

Step 9.1

Rewrite $(\sqrt[3]{5 a})^{2}$ as $\sqrt[3]{(5 a)^{2}}$ .

Step 9.2

Apply the product rule to $5 a$ .

Step 9.3

Square $5$ .

Step 10

Combine using the product rule for radicals.

Step 10.1

Combine the radical terms.

Step 10.2

Multiply $3$ by $25$ .

Step 11

Rearrange the denominator to $\frac{\sqrt[3]{75 a^{2}}}{5 a^{2}}$ .

Knowledge Notes:

The problem involves simplifying a cube root expression with variables. The solution requires understanding of several mathematical concepts:

Cube Root: The cube root of a number $x$ is a number $a$ such that $a^{3} = x$ . It is denoted as $\sqrt[3]{x}$ .
Rationalizing the Denominator: This process involves eliminating the cube root from the denominator of a fraction by multiplying the numerator and the denominator by an appropriate form of 1, which is typically the square of the cube root that is in the denominator.
Properties of Exponents: The solution uses properties such as $a^{m} \cdot a^{n} = a^{m + n}$ and $(a^{m})^{n} = a^{m n}$ .
Product Rule for Radicals: This rule states that $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$ , where $n$ is the index of the radical.
Algebraic Manipulation: The solution involves manipulating algebraic expressions, factoring, and simplifying fractions.

The solution steps are structured to gradually simplify the expression by extracting cube roots, rationalizing the denominator, and applying exponent rules to reach the simplest form.