Simplify the following expression: $ [ ( ( -\frac{1}{2} )^2 )^{-2} ]^{-1} = ? $
Understand the Problem
The question is asking to simplify the expression with exponents. We need to apply the power of a power rule and then simplify the resulting expression.
Answer
$\frac{1}{16}$
Answer for screen readers
$\frac{1}{16}$
Steps to Solve
-
Simplify the innermost exponent First, we need to evaluate the innermost expression $(-\frac{1}{2})^2$. A negative number squared becomes positive. $(-\frac{1}{2})^2 = (-\frac{1}{2}) \times (-\frac{1}{2}) = \frac{1}{4}$ So now the expression is $[(\frac{1}{4})^{-2}]^{-1}$.
-
Apply the power of a power rule We have exponents raised to exponents, so we multiply them. The expression now becomes $(\frac{1}{4})^{(-2 \times -1)}$. Since $-2 \times -1 = 2$, we have $(\frac{1}{4})^2$.
-
Evaluate the final expression Now we square $\frac{1}{4}$ to get the final result. $(\frac{1}{4})^2 = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$
$\frac{1}{16}$
More Information
The power of a power rule states that when you raise a power to a power, you multiply the exponents: $(a^m)^n = a^{m \times n}$
Tips
A common mistake is not accounting for the negative sign, in that students may incorrectly calculate $(-\frac{1}{2})^2$ as $-\frac{1}{4}$. Another mistake includes not multiplying the exponents correctly. Furthermore, another common mistake is forgetting the rule of multiplying a fraction by itself.
AI-generated content may contain errors. Please verify critical information
