Simplify p( square root of 2/3*(40))^2((2(40))/( square root of 3))

Solution:

Step 1:

Multiply $2$ by $40$ to get $80$. The expression becomes $p{\left(\sqrt{\frac{2}{3}\cdot 40}\right)}^2\cdot \frac{80}{\sqrt{3}}$.

Step 2:

Combine the fraction $\frac{2}{3}$ with $40$ to get $\frac{80}{3}$. The expression simplifies to $p{\left(\sqrt{\frac{80}{3}}\right)}^2\cdot \frac{80}{\sqrt{3}}$.

Step 3:

The square root of a fraction can be expressed as the fraction of square roots. Rewrite $\sqrt{\frac{80}{3}}$ as $\frac{\sqrt{80}}{\sqrt{3}}$. The expression is now $p{\left(\frac{\sqrt{80}}{\sqrt{3}}\right)}^2\cdot \frac{80}{\sqrt{3}}$.

Step 4:

Simplify the square root in the numerator by factoring $80$ into $16\cdot 5$. Since $16$ is a perfect square, it can be taken out of the square root as $4$. The expression becomes $p{\left(\frac{4\sqrt{5}}{\sqrt{3}}\right)}^2\cdot \frac{80}{\sqrt{3}}$.

Step 5:

Rationalize the denominator by multiplying $\frac{4\sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$. The expression now is $p{\left(\frac{4\sqrt{5}\sqrt{3}}{3}\right)}^2\cdot \frac{80}{\sqrt{3}}$.

Step 6:

Use the power rule to distribute the exponent over the fraction. The expression becomes $p\cdot \frac{(4\sqrt{15}{)}^2}{3^2}\cdot \frac{80}{\sqrt{3}}$.

Step 7:

Simplify the numerator by squaring $4$ to get $16$ and $\sqrt{15}$ to get $15$. The expression is now $p\cdot \frac{16\cdot 15}{9}\cdot \frac{80}{\sqrt{3}}$.

Step 8:

Combine terms by multiplying $16\cdot 15$ to get $240$ and $80$ by $p$. The expression simplifies to $\frac{240p}{9}\cdot \frac{80}{\sqrt{3}}$.

Step 9:

Rationalize the second fraction by multiplying $\frac{80}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$. The expression becomes $\frac{80p}{3}\cdot \frac{80\sqrt{3}}{3}$.

Step 10:

Finally, multiply the two fractions to get the simplified expression $\frac{6400p\sqrt{3}}{9}$.

Knowledge Notes:

1. Multiplication of Constants: Multiplying constants is a basic arithmetic operation. For example, $2\cdot 40=80$.

2. Simplifying Square Roots: The square root of a product can be expressed as the product of square roots, i.e., $\sqrt{ab}=\sqrt{a}\sqrt{b}$.

3. Rationalizing the Denominator: To rationalize a denominator containing a square root, multiply the numerator and denominator by the square root. This eliminates the square root from the denominator.

4. Power Rule: The power rule for exponents states that $(a^m{)}^n=a^{mn}$. This rule is used to distribute exponents over products and quotients.

5. Factoring Perfect Squares: Recognizing perfect squares within a number allows for simplification. For example, $80$ can be factored into $16\cdot 5$, where $16$ is a perfect square and can be square rooted to $4$.

6. Combining Like Terms: When terms are similar, they can be combined through addition or multiplication. For example, $\frac{240}{9}$ simplifies to $\frac{80}{3}$ when divided by $3$.

7. LaTeX Formatting: Mathematical expressions can be neatly formatted using LaTeX, a typesetting system used for displaying mathematical notation.