Simplify 8 fourth root of 24x^2y^3(8 fourth root of 24x^2y^3)

Solution:

Step:1

Perform the multiplication of $8 \sqrt[4]{24 x^{2} y^{3}}$ with itself.

Step:1.1

Compute the product of $8$ and $8$ to get $64 \sqrt[4]{24 x^{2} y^{3}} \sqrt[4]{24 x^{2} y^{3}}$.

Step:1.2

Express $\sqrt[4]{24 x^{2} y^{3}}$ as raised to the power $1$, resulting in $64 (\sqrt[4]{24 x^{2} y^{3}})^{1} \sqrt[4]{24 x^{2} y^{3}}$.

Step:1.3

Again, express the second $\sqrt[4]{24 x^{2} y^{3}}$ as raised to the power $1$, obtaining $64 (\sqrt[4]{24 x^{2} y^{3}})^{1} (\sqrt[4]{24 x^{2} y^{3}})^{1}$.

Step:1.4

Apply the exponent multiplication rule $a^{m} a^{n} = a^{m + n}$ to combine the roots into $64 (\sqrt[4]{24 x^{2} y^{3}})^{1 + 1}$.

Step:1.5

Sum the exponents $1$ and $1$ to simplify to $64 (\sqrt[4]{24 x^{2} y^{3}})^{2}$.

Step:2

Convert $(\sqrt[4]{24 x^{2} y^{3}})^{2}$ into $\sqrt{24 x^{2} y^{3}}$.

Step:2.1

Rewrite $\sqrt[4]{24 x^{2} y^{3}}$ as $(24 x^{2} y^{3})^{\frac{1}{4}}$ to get $64 ((24 x^{2} y^{3})^{\frac{1}{4}})^{2}$.

Step:2.2

Utilize the power rule $(a^{m})^{n} = a^{m n}$ to multiply the exponents, resulting in $64 (24 x^{2} y^{3})^{\frac{1}{4} \cdot 2}$.

Step:2.3

Multiply $\frac{1}{4}$ by $2$ to get $64 (24 x^{2} y^{3})^{\frac{2}{4}}$.

Step:2.4

Simplify the fraction $\frac{2}{4}$ by canceling common factors.

Step:2.4.1

Extract the factor of $2$ from the numerator to get $64 (24 x^{2} y^{3})^{\frac{2 \cdot 1}{4}}$.

Step:2.4.2

Proceed to cancel out common factors.

Step:2.4.2.1

Extract the factor of $2$ from the denominator, leading to $64 (24 x^{2} y^{3})^{\frac{2 \cdot 1}{2 \cdot 2}}$.

Step:2.4.2.2

Cancel the common factor of $2$, simplifying to $64 (24 x^{2} y^{3})^{\frac{\cancel{2} \cdot 1}{\cancel{2} \cdot 2}}$.

Step:2.4.2.3

Rewrite the expression as $64 (24 x^{2} y^{3})^{\frac{1}{2}}$.

Step:2.5

Express $(24 x^{2} y^{3})^{\frac{1}{2}}$ as $\sqrt{24 x^{2} y^{3}}$ to get $64 \sqrt{24 x^{2} y^{3}}$.

Step:3

Decompose $24 x^{2} y^{3}$ into $(2 x y)^{2} \cdot (6 y)$.

Step:3.1

Factor out $4$ from $24$ to obtain $64 \sqrt{4 \cdot 6 x^{2} y^{3}}$.

Step:3.2

Express $4$ as $2^{2}$, leading to $64 \sqrt{2^{2} \cdot 6 x^{2} y^{3}}$.

Step:3.3

Factor out $y^{2}$ to get $64 \sqrt{2^{2} \cdot 6 x^{2} (y^{2} y)}$.

Step:3.4

Rearrange to place $6$ next to $y$, resulting in $64 \sqrt{2^{2} x^{2} y^{2} \cdot 6 y}$.

Step:3.5

Rewrite $2^{2} x^{2} y^{2}$ as $(2 x y)^{2}$ to get $64 \sqrt{(2 x y)^{2} \cdot 6 y}$.

Step:3.6

Enclose $6 y$ in parentheses to clarify the expression as $64 \sqrt{(2 x y)^{2} \cdot (6 y)}$.

Step:4

Extract terms from under the radical, resulting in $64 (2 x y \sqrt{6 y})$.

Step:5

Multiply $2$ by $64$ to finalize the expression as $128 x y \sqrt{6 y}$.

Knowledge Notes:

The problem-solving process involves simplifying a mathematical expression that contains radicals and exponents. The key knowledge points covered in this process include:

1. Multiplication of similar terms: When multiplying two identical expressions, you can combine them by adding their exponents if they have the same base.

2. Power rule for exponents: $a^{m} a^{n} = a^{m + n}$, which is used to combine exponents with the same base.

3. Radical to exponent conversion: $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$, which allows us to rewrite radicals as exponents for easier manipulation.

4. Exponent multiplication rule: $(a^{m})^{n} = a^{m n}$, which is applied when raising a power to another power.

5. Simplifying fractions in exponents: Common factors in the numerator and denominator can be canceled to simplify the expression.

6. Square root simplification: $\sqrt{a^{2}} = a$ when $a$ is positive, which is used to pull terms out from under the radical.

7. Multiplication of coefficients: Coefficients outside the radical can be multiplied directly.

These concepts are fundamental in algebra and are often used in simplifying expressions, solving equations, and performing algebraic manipulations.