Simplify ((x^2y)^-4)/((xy)^-3)
In this problem, you are asked to simplify a given mathematical expression that involves exponents. The expression provided has two parts, with the first part being ((x^2y)^-4) and the second part being ((xy)^-3). The caret symbol (^) denotes an exponent, and the negative sign in front of the exponent signifies that you're dealing with the reciprocal of the base raised to a positive exponent. To simplify the expression, you would typically need to apply the rules of exponents, such as the power rule (which states that when you raise a power to a power, you multiply the exponents), the product rule (which tells you that when you multiply two powers with the same base, you add the exponents), and the rule for dealing with negative exponents (which states that you can take the reciprocal of the base and change the negative exponent to a positive exponent). The goal is to combine and reduce the expression to its simplest form.
$\frac{\left(\left(\right. x^{2} y \left.\right)\right)^{- 4}}{\left(\left(\right. x y \left.\right)\right)^{- 3}}$
Answer
Solution:
Step 1: Simplify the given expression.
Step 1.1
Utilize the rule for negative exponents, $b^{-n} = \frac{1}{b^n}$, to reposition $\left(x^2y\right)^{-4}$ to the denominator.
$$\frac{1}{\left(xy\right)^{-3}\left(x^2y\right)^4}$$
Step 1.2
Apply the negative exponent rule, $\frac{1}{b^{-n}} = b^n$, to move $\left(xy\right)^{-3}$ to the numerator.
$$\frac{\left(xy\right)^3}{\left(x^2y\right)^4}$$
Step 1.3
Invoke the product rule for exponents on $xy$.
$$\frac{x^3y^3}{\left(x^2y\right)^4}$$
Step 2: Simplify the denominator.
Step 2.1
Apply the product rule to $x^2y$.
$$\frac{x^3y^3}{\left(x^2\right)^4y^4}$$
Step 2.2
Handle the exponents in $\left(x^2\right)^4$.
Step 2.2.1
Use the power rule for exponents, $\left(a^m\right)^n = a^{mn}$.
$$\frac{x^3y^3}{x^{2 \cdot 4}y^4}$$
Step 2.2.2
Perform the multiplication of $2$ by $4$.
$$\frac{x^3y^3}{x^8y^4}$$
Step 3: Reduce the expression by eliminating common factors.
Step 3.1
Eliminate the common $x^3$ factor from $x^3$ and $x^8$.
Step 3.1.1
Extract $x^3$ from $x^3y^3$.
$$\frac{x^3(y^3)}{x^8y^4}$$
Step 3.1.2
Remove the common factors.
Step 3.1.2.1
Factor out $x^3$ from $x^8y^4$.
$$\frac{x^3(y^3)}{x^3(x^5y^4)}$$
Step 3.1.2.2
Eliminate the shared factor.
$$\frac{\cancel{x^3}y^3}{\cancel{x^3}(x^5y^4)}$$
Step 3.1.2.3
Rewrite the simplified expression.
$$\frac{y^3}{x^5y^4}$$
Step 3.2
Remove the common $y^3$ factor from $y^3$ and $y^4$.
Step 3.2.1
Introduce a multiplicative identity, $1$.
$$\frac{y^3 \cdot 1}{x^5y^4}$$
Step 3.2.2
Cancel out the common factors.
Step 3.2.2.1
Factor out $y^3$ from $x^5y^4$.
$$\frac{y^3 \cdot 1}{y^3(x^5y)}$$
Step 3.2.2.2
Eliminate the shared factor.
$$\frac{\cancel{y^3} \cdot 1}{\cancel{y^3}(x^5y)}$$
Step 3.2.2.3
Present the final simplified expression.
$$\frac{1}{x^5y}$$
Knowledge Notes:
Negative Exponent Rule: The negative exponent rule states that for any nonzero number $b$ and any integer $n$, $b^{-n} = \frac{1}{b^n}$. This rule is used to transform expressions with negative exponents into their reciprocal forms with positive exponents.
Product Rule for Exponents: The product rule for exponents says that when multiplying two powers that have the same base, you can add the exponents: $a^m \cdot a^n = a^{m+n}$.
Power Rule for Exponents: The power rule for exponents indicates that when raising a power to another power, you multiply the exponents: $\left(a^m\right)^n = a^{mn}$.
Simplifying Expressions: Simplifying expressions often involves applying exponent rules, combining like terms, and reducing fractions by eliminating common factors in the numerator and denominator.
Common Factors: When simplifying fractions, if the numerator and denominator share a common factor, it can be cancelled out to simplify the expression further. This is based on the principle that any number divided by itself equals one.
