(78)−4 as a fraction raised to a positive exponent.
(78)−4=
The Correct Answer and Explanation is :
To express ((78)^{-4}) as a fraction raised to a positive exponent, we can use the properties of exponents. The negative exponent indicates that we take the reciprocal of the base and raise it to the positive of that exponent.
Step-by-Step Explanation
- Understanding Negative Exponents: The rule for negative exponents states that (a^{-n} = \frac{1}{a^n}). This means if we have a negative exponent, we can rewrite the expression as a fraction with the base in the denominator.
- Applying the Rule: For our expression ((78)^{-4}), we can apply the negative exponent rule: [
(78)^{-4} = \frac{1}{(78)^4}
] - Finding the Positive Exponent: Now that we have rewritten the expression, we see that ((78)^{-4}) is equivalent to (\frac{1}{(78)^4}). Here, ((78)^4) is a positive exponent because the exponent itself is now positive.
- Understanding the Fraction: The expression (\frac{1}{(78)^4}) represents a fraction where the numerator is 1 and the denominator is (78) raised to the fourth power. This shows that the value of the fraction decreases as the base (78) is raised to a higher exponent.
- Calculating the Value (optional): If we wanted to compute ((78)^4), we could find that: [
(78)^4 = 78 \times 78 \times 78 \times 78
] However, for our purposes, simply expressing it as a fraction suffices. The specific numeric value isn’t necessary to understand the exponent transformation.
Conclusion
Thus, the expression ((78)^{-4}) can be expressed as a fraction with a positive exponent:
[
(78)^{-4} = \frac{1}{(78)^4}
]
This demonstrates the fundamental properties of exponents and the simplification process used in algebra to handle negative exponents effectively.