Simplify the expression.
$\begin{array}{l}
\left(\frac{x^4}{y}\right)^2 \\
\frac{x^{[?]}}{y}
\end{array}$
Answer (1)
Apply the power of a quotient rule: ( y x 4 ) 2 = y 2 ( x 4 ) 2 .
Apply the power of a power rule: ( x 4 ) 2 = x 4 × 2 = x 8 .
Simplify the expression: y 2 x 8 .
The exponent of x in the simplified expression is 8 .
Explanation
Understanding the Problem We are given the expression ( y x 4 ) 2 and asked to simplify it.
Applying the Power of a Quotient Rule To simplify the expression, we need to apply the power of a quotient rule, which states that ( b a ) n = b n a n . Applying this rule to our expression, we get: ( y x 4 ) 2 = y 2 ( x 4 ) 2
Applying the Power of a Power Rule Next, we need to apply the power of a power rule, which states that ( a m ) n = a m × n . Applying this rule to the numerator, we get: ( x 4 ) 2 = x 4 × 2 = x 8 So our expression becomes: y 2 x 8
Simplified Expression Therefore, the simplified expression is y 2 x 8 .
Finding the Exponent The question asks for the expression in the form y x [ ?] . However, the simplified expression we found is y 2 x 8 . There seems to be a typo in the original question, as the denominator should be y 2 instead of y . Assuming the question meant to ask for the exponent of x in the simplified expression, the exponent is 8.
Examples
Understanding how to simplify expressions with exponents is crucial in many areas, such as calculating areas and volumes. For example, if you have a square with side length x 2 , its area is ( x 2 ) 2 = x 4 . Similarly, in physics, understanding exponents is essential for calculations involving quantities like energy and momentum.
