Answer :
To solve the expression [tex]\(\left(-\frac{1}{2}\right)^4\)[/tex], we need to raise the base [tex]\(\left(-\frac{1}{2}\right)\)[/tex] to the power of 4. Here are the detailed steps:
1. Base and Exponent Identification:
- The base is [tex]\(-\frac{1}{2}\)[/tex].
- The exponent is 4.
2. Understanding the Exponentiation:
- When raising a fraction to a power, both the numerator and the denominator are raised to that power.
- Additionally, since the exponent is even, any negative sign in the base will be eliminated because a negative number raised to an even power results in a positive number.
3. Raise the Numerator to the Power:
- The numerator of the base is [tex]\(-1\)[/tex].
- [tex]\((-1)^4 = (-1) \times (-1) \times (-1) \times (-1) = 1\)[/tex].
4. Raise the Denominator to the Power:
- The denominator of the base is [tex]\(2\)[/tex].
- [tex]\(2^4 = 2 \times 2 \times 2 \times 2 = 16\)[/tex].
5. Combine the Results:
- Combine the results from the numerator and the denominator:
[tex]\[ \left(-\frac{1}{2}\right)^4 = \frac{(-1)^4}{2^4} = \frac{1}{16}. \][/tex]
6. Simplify the Result:
- The fraction [tex]\(\frac{1}{16}\)[/tex] is in its simplest form.
So, the value of [tex]\(\left(-\frac{1}{2}\right)^4\)[/tex] is [tex]\(\boxed{0.0625}\)[/tex].
1. Base and Exponent Identification:
- The base is [tex]\(-\frac{1}{2}\)[/tex].
- The exponent is 4.
2. Understanding the Exponentiation:
- When raising a fraction to a power, both the numerator and the denominator are raised to that power.
- Additionally, since the exponent is even, any negative sign in the base will be eliminated because a negative number raised to an even power results in a positive number.
3. Raise the Numerator to the Power:
- The numerator of the base is [tex]\(-1\)[/tex].
- [tex]\((-1)^4 = (-1) \times (-1) \times (-1) \times (-1) = 1\)[/tex].
4. Raise the Denominator to the Power:
- The denominator of the base is [tex]\(2\)[/tex].
- [tex]\(2^4 = 2 \times 2 \times 2 \times 2 = 16\)[/tex].
5. Combine the Results:
- Combine the results from the numerator and the denominator:
[tex]\[ \left(-\frac{1}{2}\right)^4 = \frac{(-1)^4}{2^4} = \frac{1}{16}. \][/tex]
6. Simplify the Result:
- The fraction [tex]\(\frac{1}{16}\)[/tex] is in its simplest form.
So, the value of [tex]\(\left(-\frac{1}{2}\right)^4\)[/tex] is [tex]\(\boxed{0.0625}\)[/tex].
