Simplify ((6x^4)/(5y))^-2
The question is asking for the simplification of a given mathematical expression that involves exponentiation and algebraic terms. The expression is presented in fractional form with a negative exponent, which means that you would need to apply exponent rules to simplify it. The expression contains variables raised to a power, and it is inverted due to the negative exponent. You're expected to manipulate the expression to make it simpler and possibly to rewrite it without a negative exponent.
$\left(\left(\right. \frac{6 x^{4}}{5 y} \left.\right)\right)^{- 2}$
Answer
Solution:
Step 1
Invert the base and change the negative exponent to a positive one: $\left(\frac{6x^4}{5y}\right)^{-2} = \left(\frac{5y}{6x^4}\right)^{2}$
Step 2
Distribute the exponent over the fraction using the power of a product rule: $\left(\frac{ab}{cd}\right)^n = \frac{a^n b^n}{c^n d^n}$
Step 2.1
Raise both the numerator and the denominator inside the fraction to the power of 2: $\left(\frac{5y}{6x^4}\right)^2 = \frac{(5y)^2}{(6x^4)^2}$
Step 2.2
Apply the exponent to each factor in the numerator: $\frac{(5^2)(y^2)}{(6x^4)^2}$
Step 2.3
Apply the exponent to each factor in the denominator: $\frac{(5^2)(y^2)}{(6^2)(x^4)^2}$
Step 3
Calculate the square of 5: $\frac{25y^2}{(6^2)(x^4)^2}$
Step 4
Simplify the expression by working on the denominator.
Step 4.1
Calculate the square of 6: $\frac{25y^2}{36(x^4)^2}$
Step 4.2
Apply the power of a power rule to the term with the exponent in the denominator.
Step 4.2.1
Use the rule $(a^m)^n = a^{mn}$ to simplify the expression: $\frac{25y^2}{36x^{4 \cdot 2}}$
Step 4.2.2
Multiply the exponents for $x$: $\frac{25y^2}{36x^8}$
Knowledge Notes:
To solve the given problem, several algebraic rules are applied:
Negative Exponent Rule: For any non-zero number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$. This rule is used to convert a negative exponent into a positive one by taking the reciprocal of the base.
Power of a Product Rule: When raising a product to an exponent, the exponent applies to each factor individually. That is, $(ab)^n = a^n b^n$.
Power of a Quotient Rule: Similarly, when raising a quotient to an exponent, the exponent applies to both the numerator and the denominator: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.
Power of a Power Rule: When an exponent is raised to another exponent, you multiply the exponents: $(a^m)^n = a^{mn}$.
Simplification of Numerical Exponents: When you have a number raised to an exponent, you calculate the value by multiplying the number by itself as many times as the exponent indicates. For example, $5^2 = 5 \times 5 = 25$.
Multiplication of Exponents: When you have the same base with exponents being multiplied, you add the exponents: $a^m \cdot a^n = a^{m+n}$.
By applying these rules systematically, the original expression can be simplified step by step.
