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

Solution:

Step:1

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

Step:1.1

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

Step:1.2

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

Step:1.3

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

Step:1.4

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

Step:1.5

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

Step:2

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

Step:2.1

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

Step:2.2

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

Step:2.3

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

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 $64(24x^2{y}^3{)}^{\frac{2\cdot 1}{4}}$.

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 $64(24x^2{y}^3{)}^{\frac{2\cdot 1}{2\cdot 2}}$.

Step:2.4.2.2

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

Step:2.4.2.3

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

Step:2.5

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

Step:3

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

Step:3.1

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

Step:3.2

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

Step:3.3

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

Step:3.4

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

Step:3.5

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

Step:3.6

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

Step:4

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

Step:5

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

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: $(a^m{)}^n=a^{mn}$, 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.