$\frac{x^4}{x^{-2}}$ is equivalent to $x$ to the power of what?
Understand the Problem
The question asks to simplify the expression $\frac{x^4}{x^{-2}}$ and determine the exponent of $x$ in the simplified form. We need to use the rules of exponents to simplify the expression.
Answer
$6$
Answer for screen readers
The exponent of $x$ in the simplified form of $\frac{x^4}{x^{-2}}$ is $6$.
Steps to Solve
- Apply the quotient rule for exponents
The quotient rule states that when dividing exponential expressions with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$. In this case, we have $\frac{x^4}{x^{-2}}$, so we subtract the exponents: $4 - (-2)$.
- Simplify the exponent
Subtracting a negative number is the same as adding the positive number: $4 - (-2) = 4 + 2 = 6$.
- Write the simplified expression
Therefore, $\frac{x^4}{x^{-2}} = x^6$. The exponent of $x$ in the simplified form is 6.
The exponent of $x$ in the simplified form of $\frac{x^4}{x^{-2}}$ is $6$.
More Information
The expression $\frac{x^4}{x^{-2}}$ simplifies to $x^6$. This is because dividing by $x^{-2}$ is the same as multiplying by $x^2$.
Tips
A common mistake is to incorrectly apply the quotient rule and add the exponents instead of subtracting them, or to mishandle the negative sign in the exponent. Remember that $\frac{x^4}{x^{-2}} = x^{4 - (-2)} = x^{4+2} = x^6$, not $x^{4-2} = x^2$.
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