Simplify ( cube root of -7x^6y^4)/( cube root of 448y^3)

Solution:

Step 1:

Merge the cube roots $\sqrt[3]{-7x^6y^4}$ and $\sqrt[3]{448y^3}$ into a single cube root expression: $\sqrt[3]{\frac{-7x^6y^4}{448y^3}}$.

Step 2:

Simplify the fraction within the cube root by eliminating common factors.

Step 2.1:

Extract the factor of $7$ from the numerator: $\sqrt[3]{\frac{7(-x^6y^4)}{448y^3}}$.

Step 2.2:

Extract the factor of $7$ from the denominator: $\sqrt[3]{\frac{7(-x^6y^4)}{7(64y^3)}}$.

Step 2.3:

Eliminate the common factor of $7$: $\sqrt[3]{\frac{\cancel{7}(-x^6y^4)}{\cancel{7}(64y^3)}}$.

Step 2.4:

Reformulate the expression: $\sqrt[3]{\frac{-x^6y^4}{64y^3}}$.

Step 3:

Further reduce the expression by canceling out common $y$ terms.

Step 3.1:

Factor out $y^3$ from the numerator: $\sqrt[3]{\frac{y^3(-x^6y)}{64y^3}}$.

Step 3.2:

Proceed to cancel the common $y^3$ factors.

Step 3.2.1:

Factor $y^3$ from the denominator: $\sqrt[3]{\frac{y^3(-x^6y)}{y^3 \cdot 64}}$.

Step 3.2.2:

Cancel out the common $y^3$ factor: $\sqrt[3]{\frac{\cancel{y^3}(-x^6y)}{\cancel{y^3} \cdot 64}}$.

Step 3.2.3:

Rephrase the simplified expression: $\sqrt[3]{\frac{-x^6y}{64}}$.

Step 4:

Position the negative sign in front of the cube root: $\sqrt[3]{-\frac{x^6y}{64}}$.

Step 5:

Express $-\frac{x^6y}{64}$ as $((-1)^3)(\frac{x^2}{4})^3y$.

Step 5.1:

Represent $-1$ as $(-1)^3$: $\sqrt[3]{(-1)^3\frac{x^6y}{64}}$.

Step 5.2:

Extract the cube of $x^2$ from $x^6y$: $\sqrt[3]{(-1)^3\frac{(x^2)^3y}{64}}$.

Step 5.3:

Extract the cube of $4$ from $64$: $\sqrt[3]{(-1)^3\frac{(x^2)^3y}{4^3 \cdot 1}}$.

Step 5.4:

Rearrange the fraction $\frac{(x^2)^3y}{4^3 \cdot 1}$: $\sqrt[3]{(-1)^3((\frac{x^2}{4})^3y)}$.

Step 5.5:

Combine $(-1)^3$ and $(\frac{x^2}{4})^3$ into a single term: $\sqrt[3]{(-\frac{x^2}{4})^3y}$.

Step 6:

Extract terms from under the cube root: $-\frac{x^2}{4}\sqrt[3]{y}$.

Step 7:

Combine the cube root of $y$ with the fraction $\frac{x^2}{4}$: $-\frac{\sqrt[3]{y}x^2}{4}$.

Knowledge Notes:

The problem-solving process involves simplifying a cube root expression. The steps taken include combining cube roots, factoring out common terms, canceling common factors, and extracting terms from under the cube root. The process requires knowledge of the properties of radicals, factoring, and simplifying algebraic expressions.

Relevant knowledge points include:

1. Cube root simplification: $\sqrt[3]{a^3} = a$.

2. Factoring: Finding common factors in the numerator and denominator to simplify fractions.

3. Properties of exponents: $(a^m)^n = a^{mn}$ and $a^m \cdot a^n = a^{m+n}$.

4. Negative exponents and radicals: $(-a)^n = a^n$ if $n$ is even, and $(-a)^n = -a^n$ if $n$ is odd.

5. Simplifying expressions: Combining like terms and reducing fractions to their simplest form.