Simplify p( square root of 2/3*(40))^2((2(40))/( square root of 3))
The problem provided involves simplifying a mathematical expression that contains roots, fractions, and powers. The expression is a product of two major components:
The first part is the expression $p( square root of 2/3*(40))^2$, where:
$p$could be a variable or a function (not specified in the question).
The square root of $2/3$is to be multiplied by $40$.
The resulting product is then squared.
The second part of the product is $(2(40))/( square root of 3)$, where:
$40$is being multiplied by $2$.
The product is then divided by the square root of $3$.
The task is to simplify this expression by applying algebraic rules, including the distributive property, associative property, and the properties of square roots and powers, to combine the terms in such a way that the expression is in a reduced or simplest form.
$p \left(\left(\right. \sqrt{\frac{2}{3} \cdot \left(\right. 40 \left.\right)} \left.\right)\right)^{2} \left(\right. \frac{2 \left(\right. 40 \left.\right)}{\sqrt{3}} \left.\right)$
Answer
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:
Multiplication of Constants: Multiplying constants is a basic arithmetic operation. For example, $2 \cdot 40 = 80$.
Simplifying Square Roots: The square root of a product can be expressed as the product of square roots, i.e., $\sqrt{ab} = \sqrt{a}\sqrt{b}$.
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.
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.
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 $4$.
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 $3$.
LaTeX Formatting: Mathematical expressions can be neatly formatted using LaTeX, a typesetting system used for displaying mathematical notation.
